GATE PI 2021

This problem involves finding an initial value before a percentage increase over time, which is a common application of compound growth principles. We need to find the population 2 years ago ($P_0$) given the current population ($P = 11,02,500$), an annual growth rate ($r = 5\%$), and the time period ($n = 2$ years).

The standard formula for compound growth is:
$P = P_0 \left(1 + \frac{r}{100}\right)^n$
To find the population 2 years ago ($P_0$), we rearrange the formula:
$P_0 = \frac{P}{\left(1 + \frac{r}{100}\right)^n}$
Now, let's plug in the values and calculate:
First, convert the percentage rate to a decimal: $\frac{5}{100} = 0.05$.
Next, add 1 to the decimal rate: $1 + 0.05 = 1.05$.
Then, calculate the growth factor for 2 years: $(1.05)^2 = 1.05 \times 1.05 = 1.1025$.
Finally, divide the current population by this growth factor to find $P_0$:
$P_0 = \frac{11,02,500}{1.1025}$
$P_0 = 10,00,000$
The population of the city 2 years ago was $10,00,000$.

This problem tests our ability to manipulate algebraic expressions involving fractions. We are given $p$ and $q$ are positive integers and the equation:
$\frac{p}{q} + \frac{q}{p} = 3$
Our goal is to find the value of:
$\frac{p^2}{q^2} + \frac{q^2}{p^2}$
Let's simplify by substituting $x = \frac{p}{q}$. This means $\frac{1}{x} = \frac{q}{p}$.
The given equation transforms to:
$x + \frac{1}{x} = 3$
The expression we need to find becomes:
$\left(\frac{p}{q}\right)^2 + \left(\frac{q}{p}\right)^2 = x^2 + \left(\frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2}$
We can use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$. Let $a = x$ and $b = \frac{1}{x}$:
$\left(x + \frac{1}{x}\right)^2 = x^2 + \left(\frac{1}{x}\right)^2 + 2(x)\left(\frac{1}{x}\right)$
Simplify the term $2(x)\left(\frac{1}{x}\right)$:
$2(x)\left(\frac{1}{x}\right) = 2 \times \frac{x}{x} = 2 \times 1 = 2$
So, the identity becomes:
$\left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2$
We know that $x + \frac{1}{x} = 3$. Substitute this value into the equation:
$(3)^2 = x^2 + \frac{1}{x^2} + 2$
$9 = x^2 + \frac{1}{x^2} + 2$
To find $x^2 + \frac{1}{x^2}$, subtract 2 from both sides:
$x^2 + \frac{1}{x^2} = 9 - 2$
$x^2 + \frac{1}{x^2} = 7$
Since $x^2 + \frac{1}{x^2} = \frac{p^2}{q^2} + \frac{q^2}{p^2}$, the value we are looking for is 7.

To make the line $P-Q$ a line of symmetry, the figure must be identical on both sides of the line. We need to add squares to one side such that it mirrors the existing squares on the other side. By visually inspecting the given figure and adding squares to complete this mirror image, we find that a total of $6$ squares must be added to achieve symmetry.

This analogy question tests our understanding of relationships between words. The core relationship here is one of temporal direction.

1. Analyze the given pair:

   • Nostalgia is a feeling focused on the past. We experience nostalgia for something that has already happened.
   • Anticipation is a feeling focused on the future. We anticipate something that is going to happen.
   • So, the relationship is: Past-oriented emotion : Future-oriented emotion.

2. Evaluate the options based on this temporal direction:

   • A. Present : Past $-$ This doesn't match the "past : future" directional flow.
   • B. Future : Past $-$ This is the reverse of our required "past : future" flow.
   • C. Past : Future $-$ This directly mirrors the "past-oriented : future-oriented" relationship established by Nostalgia : Anticipation.
   • D. Future : Present $-$ This doesn't match the "past : future" directional flow.

3. Conclusion: The option that maintains the same logical relation (Past-oriented : Future-oriented) is C. Past, future. Therefore, Nostalgia is to anticipation as Past is to Future.

To identify the grammatically correct sentences, we need to understand the principal forms of the verb 'wake': Base Form: wake; Past Simple: woke; Past Participle: woken.

• Sentence (i) "I woke up from sleep." uses 'woke', which is the correct simple past tense of 'wake', making it grammatically correct.
• Sentence (ii) "I woked up from sleep." uses 'woked', an incorrect past tense form; the correct form is 'woke', so this sentence is grammatically incorrect.
• Sentence (iii) "I was woken up from sleep." correctly uses 'was woken', which is the past tense passive voice structure with the correct past participle 'woken', making it grammatically correct.
• Sentence (iv) "I was wokened up from sleep." uses 'wokened', which is an incorrect past participle form; the correct form is 'woken', so this sentence is grammatically incorrect.

Thus, sentences (i) and (iii) are grammatically correct.

This question tests our deductive reasoning skills in analyzing categorical statements.

The statements provided are:

1. All purple are green. (\text{P} \subseteq \text{G}$)
2. All black are green. (\text{B} \subseteq \text{G}$)

These statements mean that the entire set of 'purple' is a subset of 'green', and the entire set of 'black' is also a subset of 'green'. However, these statements do not define the relationship between 'purple' and 'black'.

Now, let's look at the conclusions:
I. Some black are purple. (\text{B} \cap \text{P} \neq \emptyset$) II. No black is purple. (\text{B} \cap \text{P} = \emptyset$)

Conclusion I suggests that there is an overlap between 'black' and 'purple'. Conclusion II suggests that 'black' and 'purple' are entirely separate.
From the given statements, both scenarios (overlap or no overlap between 'black' and 'purple' within 'green') are possible. We cannot definitively say that 'Some black are purple' is true, nor can we definitively say that 'No black is purple' is true based only on the premises.

However, Conclusion I and Conclusion II are contradictory statements. They cannot both be true simultaneously, and they cannot both be false simultaneously. If one is false, the other must be true. Since the premises allow for either possibility (overlap or no overlap), and one of these two contradictory conclusions must hold true, the logically correct option is that "Either conclusion I or II is correct."

Let's break down each statement to see if it can be concluded from the passage.

Statement (i): Nowadays, computers are present in almost all places.
The passage clearly states, "Computers are ubiquitous," and "They are used to improve efficiency in almost all fields from agriculture to space exploration." "Ubiquitous" means present everywhere, and "almost all fields" supports this widespread presence. So, statement (i) is a valid deduction.

Statement (ii): Computers cannot be used for solving problems in engineering.
The passage mentions computers are used in "almost all fields from agriculture to space exploration." Space exploration heavily involves complex engineering. Therefore, computers can be used in engineering, which contradicts statement (ii). So, statement (ii) is not a valid deduction.

Statement (iii): For humans, there are both positive and negative effects of using computers.
The passage states, "For humans, sitting in front of a computer for long hours can lead to health issues," which is a negative effect. The passage also mentions computers "improve efficiency in almost all fields," implying positive outcomes from their use. Thus, both positive and negative effects are implied. So, statement (iii) is a valid deduction.

Statement (iv): Artificial intelligence can be done without data.
The passage defines AI: "AI enables computers to learn, given enough training data." The phrase "given enough training data" explicitly indicates that data is a requirement for AI to learn. This directly refutes statement (iv). So, statement (iv) is not a valid deduction.

Based on this analysis, only statements (i) and (iii) can be deduced from the passage.

Let's break down this geometry problem step-by-step to find the cone's volume!

First, we start with a square sheet of side 1 unit. When we cut it along its main diagonal, we get two identical right-angled triangles. Each of these triangles has two sides of length 1 unit (the original sides of the square) and the third side is the diagonal. Using the Pythagorean theorem, the diagonal length is $\sqrt{1^2 + 1^2} = \sqrt{2}$ units. So, each triangle has sides of length 1, 1, and $\sqrt{2}$ units. The two shorter edges are the sides with length 1 unit.

Next, one of these triangles is revolved about its short edge. When a right-angled triangle is revolved around one of its legs (a short edge), it forms a solid cone. The leg it revolves around becomes the height ($h$) of the cone, and the other leg becomes the radius ($r$) of the cone's base. The hypotenuse becomes the slant height. Since we are revolving about a short edge of length 1 unit, we have:
Height ($h$) = 1 unit
Radius ($r$) = 1 unit
Slant height ($l$) = $\sqrt{2}$ units

Finally, to calculate the volume of this cone, we use the formula: $V = \frac{1}{3}\pi r^2 h$.
Substituting our values for $r$ and $h$:
$V = \frac{1}{3}\pi (1)^2 (1)$
$V = \frac{1}{3}\pi (1)(1)$
$V = \frac{\pi}{3}$ cubic units.

To find the number of days where one student spent at least $10\%$ more time exercising than the other, we calculate the percentage increase with respect to the smaller value for each day.

We use the formula: Percentage Increase $= \frac{\text{Difference}}{\text{Smaller Value}} \times 100$.

• Sunday: $(\frac{65 - 55}{55}) \times 100 = 18.18\%$
• Monday: $(\frac{70 - 45}{45}) \times 100 = 55.55\%$
• Tuesday: $(\frac{65 - 55}{55}) \times 100 = 18.18\%$
• Wednesday: $(\frac{60 - 50}{50}) \times 100 = 20\%$
• Thursday: $(\frac{60 - 55}{55}) \times 100 = 9.09\%$
• Friday: $(\frac{35 - 20}{20}) \times 100 = 75\%$
• Saturday: $(\frac{60 - 50}{50}) \times 100 = 20\%$

Comparing these percentages to $10\%$, we find that on Sunday, Monday, Tuesday, Wednesday, Friday, and Saturday, the increase is $10\%$ or more. Thus, there are 6 such days.

The core concept here is the formula for the Area of an equilateral triangle: $Area = \frac{\sqrt{3}}{4} \times a^2$, where $a$ is the side length.

Calculation:
Let the side of the original large equilateral triangle be $a$. When corners are cut to form a regular hexagon, the side of this regular hexagon becomes $\frac{a}{3}$. The hexagon is formed by removing three smaller equilateral triangles from the corners. Each of these smaller triangles has a side length of $\frac{a}{3}$.

The area of the original equilateral triangle ($A_T$) is $A_T = \frac{\sqrt{3}}{4} \times a^2$.

The regular hexagon can be thought of as consisting of 6 smaller equilateral triangles, each with side length $\frac{a}{3}$.
So, the area of the regular hexagon ($A_H$) is $6 \times (\frac{\sqrt{3}}{4} \times (\frac{a}{3})^2)$.

Now, to find the ratio of the area of the regular convex hexagon to the area of the original equilateral triangle:
$\frac{A_H}{A_T} = \frac{6 \times \frac{\sqrt{3}}{4} \times \left(\frac{a}{3}\right)^2}{\frac{\sqrt{3}}{4} \times a^2}$
$\frac{A_H}{A_T} = \frac{6 \times \frac{\sqrt{3}}{4} \times \frac{a^2}{9}}{\frac{\sqrt{3}}{4} \times a^2}$
Cancel out the common terms $\frac{\sqrt{3}}{4}$ and $a^2$:
$\frac{A_H}{A_T} = \frac{6 \times \frac{1}{9}}{1} = \frac{6}{9} = \frac{2}{3}$
Thus, the ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is $\frac{2}{3}$.

This problem asks for the reliability of a product after a certain time, given its time-to-failure follows an exponential distribution. The key concept here is the reliability function for an exponential distribution, which is given by $R(t) = e^{-\lambda t}$. Here, $R(t)$ is the reliability at time $t$, $e$ is the base of the natural logarithm, and $\lambda$ is the constant failure rate.

Given:
Failure rate, $\lambda = 0.00006$ per hour.
Time, $t = 4000$ hours.

Substitute these values into the reliability formula:
$R(4000) = e^{-(0.00006 \text{ per hour} \times 4000 \text{ hours})}$
First, calculate the exponent:
$\lambda t = 0.00006 \times 4000 = 0.24$
Now, calculate the reliability:
$R(4000) = e^{-0.24}$
Using a calculator, $e^{-0.24} \approx 0.786627856$.
Rounding to four decimal places, the reliability of the product after 4000 hours of operation is approximately $0.7866$.

Concurrent engineering focuses on making the product development process efficient by considering all aspects of a product's entire life cycle simultaneously, rather than sequentially. The product life cycle includes conception, planning, design, manufacturing, distribution, use, maintenance, and end-of-life (disposal, recycling). In concurrent engineering, the product design stage is crucial because it's where fundamental product characteristics are defined, and critical decisions about materials, components, and assembly methods are made. By integrating feedback from manufacturing, marketing, service, and environmental teams during this early stage, designers proactively consider factors like ease of manufacturing, durability, maintainability, and disposability. This holistic approach at the design stage helps prevent costly issues later, ensuring that all elements of the product life cycle, from conception to disposal, are addressed upfront.

The Operating Characteristic (OC) curve shows the probability of accepting a lot ($P_a$) against the proportion of defective items ($p$) in that lot. For a single sampling plan with an acceptance number ($c$) of zero, meaning the lot is accepted only if no defectives are found in the sample, and a sample size ($n$) of $n \le 10$, the probability of acceptance is given by:
$P_a = (1-p)^n$
Let's analyze the behavior of this function for $0 \le p \le 1$:
At $p=0$ (no defectives), $P_a = (1-0)^n = 1^n = 1$. The curve starts at (0, 1).
As $p$ increases, $(1-p)$ decreases, causing $P_a$ to decrease.
At $p=1$ (all defectives), $P_a = (1-1)^n = 0^n = 0$ (assuming $n>0$). The curve ends at (1, 0).
To determine the exact shape, we look at the second derivative:
$\frac{dP_a}{dp} = -n(1-p)^{n-1}$
$\frac{d^2P_a}{dp^2} = n(n-1)(1-p)^{n-2}$
If $n=1$, the second derivative is $1(0)(1-p)^{-1} = 0$, indicating a straight line ($P_a = 1-p$), which is considered convex.
If $n > 1$ (and $n \le 10$), $n(n-1)$ is positive, and $(1-p)^{n-2}$ is positive for $0 \le p < 1$. Thus, $\frac{d^2P_a}{dp^2}$ is positive, which means the function is convex. Therefore, the OC curve is a convex function.

Computerized layout design techniques arrange facilities for efficiency and cost reduction using algorithms. These algorithms are broadly categorized into Construction (builds layouts from scratch) and Improvement (refines existing layouts). Heuristic algorithms provide good solutions efficiently, though not always optimal.

Let's evaluate the options:

• Systematic Layout Planning (SLP): This is primarily a manual, qualitative procedure focusing on analyzing relationships, not a computerized improvement heuristic.
• Computerized Relative Allocation of Facilities Technique (CRAFT): CRAFT is a classic improvement type heuristic. It starts with an initial layout and iteratively swaps department locations to minimize total material handling cost. The process continues until no further cost reduction is possible from swaps. This iterative refinement makes it an improvement heuristic. The objective function it typically minimizes is:
  $\text{Total Material Handling Cost} = \sum (\text{Flow between departments}) \times (\text{Distance between departments})$
• Computerized Relationship Layout Planning (CORELAP): CORELAP is a construction heuristic. It builds a layout block-by-block based on relationship scores, rather than improving an existing one.
• Plant Layout Analysis and Evaluation Technique (PLANET): PLANET is more for evaluating and comparing layouts, not an improvement heuristic itself.

Therefore, CRAFT fits the description of an improvement type heuristic algorithm for computerized layout design.

This question tests your knowledge of common forecast error metrics. Forecast error measures quantify how much our predictions ($F_i$) deviate from the actual observed values ($A_i$).

Let's examine each option:

• A. Mean Absolute Deviation (MAD): This is a standard measure. It calculates the average of the absolute differences between actual and forecasted values. Its formula is:
  $\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |A_i - F_i|$
  MAD gives the average magnitude of the error.

• B. Mean Squared Error (MSE): This is also a widely used metric. It calculates the average of the squared differences between actual and forecasted values. The formula is:
  $\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (A_i - F_i)^2$
  MSE penalizes larger errors more heavily because of the squaring.

• C. Mean Absolute Percent Error (MAPE): This metric expresses the error as a percentage of the actual value, making it useful for comparing forecast accuracy across different scales. The formula is:
  $\text{MAPE} = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{A_i - F_i}{A_i} \right| \times 100\%$
  Be careful when $A_i$ is zero or very close to zero.

• D. Mean Sum Product Error (MSPE): This term is not a standard or recognized measure of forecast error in the literature. While one could theoretically construct such a calculation, it is not used to evaluate forecast performance like MAD, MSE, or MAPE.

Therefore, the option that is NOT a measure of forecast error is Mean Sum Product Error (MSPE).

Pearlite is a layered microstructure formed in steel, especially in eutectoid steel (which has about 0.77% carbon). When austenite, a high-temperature phase, is slowly cooled below the eutectoid temperature (around $727^\circ\text{C}$), it undergoes a transformation called the eutectoid reaction:

$\text{Austenite} \xrightarrow{\text{Cooling below } 727^\circ\text{C}} \alpha\text{-ferrite} + \text{Cementite}$

This reaction produces pearlite, which consists of alternating layers of two distinct phases: $\alpha$-ferrite (a soft, ductile phase with very low carbon solubility, max $0.022\%$ at $727^\circ\text{C}$) and cementite (\text{Fe}_3\text{C}$, a hard, brittle intermetallic compound containing$6.67%$carbon). Therefore, pearlite \in eutectoid steel is composed of alternating layers of$\alpha$-ferrite and Cementite.

Let's break down these defects to see which one doesn't fit with welding.

• Angular Distortion: This is a very common welding defect. It happens when the intense, localized heat from welding causes parts of the metal to expand and contract unevenly, leading to the workpiece bending or warping out of its original shape.
• Hot Tear: Also known as hot cracking, this is a type of crack that forms during the welding process while the metal is still cooling down from its molten state. It occurs because the metal isn't ductile enough at high temperatures to handle the stresses from contraction.
• Hydrogen Embrittlement: This defect primarily affects high-strength steels. During welding, if hydrogen (often from moisture) gets absorbed into the metal, it makes the material brittle and prone to cracking under stress. It's a significant welding-related issue.
• Earring: This defect is NOT associated with welding. Earring occurs in sheet metal forming processes, like deep drawing, where uneven "ears" or wavy edges form on the part due to the material's directional properties (anisotropy) during stretching.

Therefore, Earring is the defect that is not typically linked to welding processes.

This question tests your knowledge of common manufacturing processes and their suitable applications for various engineering components. We need to match each component to its most appropriate manufacturing method:

• P: Aluminum Alloy Piston for IC Engine
  Pistons operate under high temperatures and pressures and often have complex internal geometries. Sand casting (3) is excellent for creating intricate shapes from aluminum alloys, making it a standard process for pistons.

• Q: Low Carbon Steel Oil Pan
  An oil pan is typically a hollow, tray-like component made from sheet metal. Deep drawing (4) is a metal forming process specifically designed to produce hollow, cup-shaped parts from flat sheet metal blanks, which is ideal for oil pans.

• R: Tungsten Carbide Cutting Tool Insert
  Tungsten carbide is an extremely hard, brittle material suitable for cutting tools. Powder metallurgy (2) is the primary method for manufacturing such materials; it involves compacting powdered materials (like tungsten carbide and a binder) and then sintering them at high temperatures to form a dense, solid part.

• S: Plastic Bottle
  Plastic bottles are hollow containers. Blow molding (1) is a widely used process for manufacturing hollow plastic parts; it involves inflating a molten plastic preform (parison) inside a mold to take its shape.

Based on these explanations, the correct matches are P-3, Q-4, R-2, S-1.

In turning operations, the relationship between cutting speed ($V$) and tool life ($T$) is described by Taylor's tool life equation: $VT^n = C$. Here, $n$ is the tool life exponent and $C$ is a constant.

We are given that when the cutting speed is doubled ($V_2 = 2V_1$), the tool life reduces to one-eighth of its original value ($T_2 = \frac{1}{8}T_1$).
Using Taylor's equation for both conditions:
$V_1 T_1^n = C$
$V_2 T_2^n = C$
Equating them: $V_1 T_1^n = V_2 T_2^n$

Substitute the given relationships:
$V_1 T_1^n = (2V_1) \left(\frac{1}{8} T_1\right)^n$
$T_1^n = 2 \left(\frac{1}{8}\right)^n T_1^n$
$1 = 2 \left(\frac{1}{8}\right)^n$
$\left(\frac{1}{8}\right)^n = \frac{1}{2}$

To solve for $n$, express both sides with a common base:
$\left(\left(\frac{1}{2}\right)^3\right)^n = \frac{1}{2}^1$
$\left(\frac{1}{2}\right)^{3n} = \left(\frac{1}{2}\right)^1$
Equating the exponents:
$3n = 1$
$n = \frac{1}{3}$

This question asks us to identify the mechanism that does NOT primarily convert rotational motion into linear (translational) motion in machine tools.

• Screw-Nut System: This system directly converts screw rotation into linear motion of the nut (or vice versa), commonly seen in lead screws of lathes for tool feed.
• Cam and Cam Follower System: A rotating cam with a specific profile pushes a follower, converting the cam's rotary motion into the follower's linear or oscillating motion, used for automated movements.
• Whitworth Mechanism: This mechanism takes rotary input and produces a reciprocating (linear) output, characterized by a quick return stroke, frequently found in shaping machines.
• 4-Bevel Gear Type Differential Mechanism: Differentials are designed to manage or distribute rotational motions, typically by splitting a single input rotation into two varying output rotations or combining two inputs into one output (e.g., in vehicle axles). Its core function is not the direct conversion of a single rotary input into a linear translational output.

Therefore, the 4-bevel gear type differential mechanism primarily deals with distributing or combining rotational motions, rather than transforming rotation into translation like the other options.

This question asks us to match various measuring features with the most appropriate instrument. Let's break down each pairing:

P. Flatness error of a surface plate: Surface plates are fundamental for precision measurements, and their flatness is crucial. To measure minute deviations from a perfect plane with extremely high accuracy (often in micrometers), an Optical interferometer (4) is used. It works by analyzing the interference patterns of light, making it ideal for such precise flatness measurements.

Q. Profile of a cam: A cam's precise shape dictates its function. To measure its profile, we need to map its contour by correlating angular position with displacement. A Dividing head and dial gauge (3) setup is perfect for this. The dividing head precisely rotates the cam, while the dial gauge measures the radial displacement at each angular position.

R. Alignment error of a machine tool slide way: The accuracy of machine tools heavily relies on the proper alignment of components like slide ways. Detecting straightness and angular alignment errors is critical. An Autocollimator (1) is an optical instrument specifically designed to check straightness and alignment over distances by projecting and measuring the angular deviation of a reflected light beam.

S. Pitch and angle errors of screw thread: Screw threads have critical geometric parameters like pitch (axial distance between threads) and thread angle. Measuring these features requires magnification and precise linear measurement. A Tool maker's microscope (2) is well-suited for this task, as it provides magnified views of the thread form and has a calibrated stage for accurate measurement of pitch and angles.

Therefore, the correct matching is P-4, Q-3, R-1, S-2. This corresponds to Option D.

In Electric Discharge Machining (EDM), the duty factor (DF) tells us the proportion of time the electrical pulse is on during one complete cycle. It's calculated as the ratio of pulse on-time ($T_{on}$) to the total pulse period ($T$).

First, let's find the total pulse period ($T$). The frequency ($f$) is given as $10 \text{ kHz}$, which is $10 \times 10^3 \text{ Hz}$.
The period is the inverse of frequency:
$T = \frac{1}{f} = \frac{1}{10 \times 10^3 \text{ Hz}} = 0.0001 \text{ s}$
To work with microseconds (µs), we convert $T$:
$T = 0.0001 \text{ s} \times \left( \frac{10^6 \:\mu\text{s}}{1 \text{ s}} \right) = 100 \:\mu\text{s}$
Now, we find the pulse on-time ($T_{on}$). We know that the total period ($T$) is the sum of pulse on-time ($T_{on}$) and pulse off-time ($T_{off}$). The pulse off-time ($T_{off}$) is given as $40 \:\mu\text{s}$.
$T_{on} = T - T_{off} = 100 \:\mu\text{s} - 40 \:\mu\text{s} = 60 \:\mu\text{s}$
Finally, we calculate the duty factor (DF):
$\text{DF} = \frac{T_{on}}{T} = \frac{60 \:\mu\text{s}}{100 \:\mu\text{s}} = 0.60$
Alternatively, the duty factor can also be calculated as:
$\text{DF} = 1 - \frac{T_{off}}{T} = 1 - \frac{40 \:\mu\text{s}}{100 \:\mu\text{s}} = 1 - 0.40 = 0.60$

The core concept here is the relationship between bending moment and beam curvature, given by the pure bending equation: $\frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$. From this, we are interested in the curvature term, $\frac{1}{R} = \frac{M}{E I}$.

Given: Uniformly distributed load $W = 10 \text{ kN/m} = 10 \times 10^3 \text{ N/m}$, beam length $L = 0.3 \text{ m}$, and flexural rigidity $E I = 5000 \text{ Nm}^2$.

First, we need to find the bending moment ($M$) at the fixed end. For a cantilever beam with a uniformly distributed load, the bending moment at a distance $x$ from the free end is $M_x = \frac{Wx^2}{2}$. The maximum bending moment occurs at the fixed end ($x=L$):
$M = \frac{WL^2}{2} = \frac{10 \times 10^3 \times (0.3)^2}{2} = 450 \text{ N-m}$
Now, we can calculate the beam curvature at the fixed end using the derived formula:
$\frac{1}{R} = \frac{M}{E I} = \frac{450}{5000} = 0.09 \text{ m}^{-1}$

The critical load ($P_{cr}$) at which a long column buckles is given by Euler's buckling formula: $P_{cr} = \frac{\pi^2EI}{L_e^2}$, where $EI$ is the flexural rigidity and $L_e$ is the effective length. For a column with both ends pinned (or hinged), the effective length $L_e$ is equal to the actual length $L$.

Given:

• Actual length, $L = 2 \text{ m}$
• Flexural rigidity, $EI = 2000 \text{ Nm}^2$

Since both ends are pinned, $L_e = L = 2 \text{ m}$.

Substituting these values into the formula:
$P_{cr} = \frac{\pi^2 \times 2000}{2^2} = \frac{\pi^2 \times 2000}{4} = 4934.8 \text{ N}$

Rounding to the nearest integer, $P_{cr} = 4935 \text{ N}$.

This problem involves calculating the contact pressure in a press-fit assembly. The core concept here is how the radial interference ($\delta$) relates to the stresses generated at the interface.

The radial interference ($\delta$) in a press-fit is given by the formula:
$\delta = r_{\text{inner}} \times \left[ \frac{(\sigma_h)_{\text{outer cylinder}} + (\sigma_h)_{\text{inner cylinder}}}{E} \right]$

For the outer cylinder, the hoop stress ($\sigma_h$) at the inner diameter ($d_i$) due to an internal pressure $P_i$ (which is our contact pressure $P$) is given by:
$\sigma_h = P_i \times \left[ \frac{d_o^2 + d_i^2}{d_o^2 - d_i^2} \right]$
Here, $d_i = 2r_i$ and $d_o = 2r_o$. So, replacing diameters with radii:
$\sigma_h = P \times \left[ \frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} \right]$
Given $r_{\text{inner}} = 25 \, \text{mm}$ and $r_{\text{outer}} = 40 \, \text{mm}$, the hoop stress in the outer cylinder at the interface is:
$\sigma_h = P \times \left[ \frac{40^2 + 25^2}{40^2 - 25^2} \right]\$ = 2.282P$
For the inner cylinder, the hoop stress ($\sigma_h$) at the outer diameter (which is the contact surface) is simply equal to the contact pressure $P$, i.e., $(\sigma_h)_{\text{inner cylinder}} = P$.

Now, substitute these stresses and the given values into the radial interference equation:
Given $\delta = 0.05 \, \text{mm}$, $r_{\text{inner}} = 25 \, \text{mm}$, $E = 200 \, \text{GPa} = 200 \times 10^3 \, \text{MPa}$:
$0.05 = 25 \times \left[ \frac{(2.282P) + P}{200 \times 10^3} \right]$
Solving for $P$:
$0.05 = 25 \times \left[ \frac{3.282P}{200 \times 10^3} \right]$
$P = \frac{0.05 \times 200 \times 10^3}{25 \times 3.282} = 121.9 \, \text{MPa}$

In fluid dynamics, dimensionless numbers help us understand fluid behavior by comparing the strengths of different forces. The question asks for the dimensionless number defined as the ratio of inertial force to viscous force. This is precisely the definition of the Reynolds number ($Re$).

The Reynolds number ($Re$) is mathematically expressed as:
$Re = \frac{\text{Inertial Force}}{\text{Viscous Force}}$
For a fluid with density $\rho$, viscosity $\mu$, mean velocity $v$, and characteristic length $L$, the formula is:
$Re = \frac{\rho v L}{\mu}$
A high Reynolds number indicates dominant inertial forces (turbulent flow), while a low Reynolds number indicates dominant viscous forces (laminar flow).

Let's quickly review the other options:

• Mach number ($Ma$) is the ratio of flow speed ($v$) to the speed of sound ($c$), $Ma = \frac{v}{c}$, used for compressible flows.
• Froude number ($Fr$) is the ratio relating inertia to gravity, $Fr = \frac{v}{\sqrt{gL}}$, important in open channel flows.
• Weber number ($We$) is the ratio of inertial force to surface tension forces ($\sigma$), $We = \frac{\rho v^2 L}{\sigma}$, used when surface tension is significant.

Therefore, based on its definition as the ratio of inertial force to viscous force, the Reynolds number is the correct answer.

This problem involves calculating the capillary rise, a common phenomenon in fluid mechanics. The height to which a liquid rises in a narrow tube due to surface tension is governed by Jurin's Law, given by:

$h = \frac{2 \gamma \cos \theta}{\rho g r}$

Here, $h$ is the capillary rise, $\gamma$ is the surface tension, $\theta$ is the contact angle, $\rho$ is the fluid density, $g$ is the acceleration due to gravity, and $r$ is the radius of the capillary tube.

Let's gather the given values and ensure they are in consistent SI units:
Inner diameter ($d$) = 3 mm = $3 \times 10^{-3}$ m
Radius ($r$) = $d/2$ = $1.5$ mm = $1.5 \times 10^{-3}$ m
Density ($\rho$) = 900 kg/m$^3$
Surface tension ($\gamma$) = 0.1 N/m
Contact angle ($\theta$) = 30$^\circ$
Acceleration due to gravity ($g$) = $9.81 \ m/s^2$

Now, substitute these values into Jurin's Law formula:
First, calculate $\cos(30^\circ) = 0.8660$.
$h = \frac{2 \times (0.1 \ N/m) \times 0.8660}{(900 \ kg/m^3) \times (9.81 \ m/s^2) \times (1.5 \times 10^{-3} \ m)}$
$h = \frac{0.1732}{13.2435}$
$h \approx 0.013078 \ m$
To match the options, convert the height to millimeters:
$h \approx 0.013078 \ m \times 1000 \ mm/m = 13.078 \ mm$

Let's understand this problem using the Von Mises yield criterion, which is particularly relevant for ductile materials like steel. This criterion states that yielding occurs when the distortion energy per unit volume under applied stress exceeds the distortion energy at the yield point in simple tension. Mathematically, for a state of stress defined by principal stresses $\sigma_1, \sigma_2, \sigma_3$, yielding begins when:

$(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \le 2 \sigma_y^2$

Given values are $\sigma_1 = \sigma$, $\sigma_2 = 0$, $\sigma_3 = -\sigma$, and the yield strength $\sigma_y = 700 \text{ MPa}$. Substituting these into the Von Mises criterion for the initiation of yielding (where the inequality becomes an equality):

$(\sigma - 0)^2 + (0 - (-\sigma))^2 + ((-\sigma) - \sigma)^2 = 2 (700)^2$
$\sigma^2 + \sigma^2 + (-2\sigma)^2 = 2 (700)^2$
$\sigma^2 + \sigma^2 + 4\sigma^2 = 2 (700)^2$
$6\sigma^2 = 2 (700)^2$
$\sigma^2 = \frac{2 (700)^2}{6} = \frac{(700)^2}{3}$
$\sigma = \frac{700}{\sqrt{3}}$
Therefore, the stress $\sigma$ at which plastic yielding initiates is:
$\sigma \approx 404.14 \text{ MPa}$

To find the total longitudinal true strain, we use the formula for true strain: $\varepsilon_T = \ln \left(\frac{L_f}{L_0}\right)$, where $L_f$ is the final length and $L_0$ is the initial length. In this problem, the specimen is extended in two stages, but we are interested in the total true strain from the initial length to the final length after both stages. The initial length $L_0 = 50$ mm and the final length after both stages is $L_f = 60$ mm (since it goes from 50 mm to 55 mm, then from 55 mm to 60 mm). Therefore, the total longitudinal true strain is calculated as:

$\varepsilon_T = \ln \left(\frac{60}{50}\right) = 0.18232$.

To find the tightening torque, we use the formula that relates it to the pretension force, bolt diameter, and torque coefficient. The general expression for tightening torque is $T = F_{pre} \times d \times K$, where $K$ is the torque coefficient, which represents $\frac{1}{2}\tan(\phi + \alpha)$.

Given:
Pretension force ($F_{pre}$) = $350$ kN = $350 \times 10^3$ N
Bolt nominal diameter ($d$) = $30$ mm = $30 \times 10^{-3}$ m
Torque coefficient ($K$) = $0.2$

Using the formula:
Tightening Torque ($T$) = $F_{pre} \times d \times K$
$T = (350 \times 10^3 \text{ N}) \times (30 \times 10^{-3} \text{ m}) \times 0.2$
$T = 2100 \text{ Nm}$

Concept: Power transmission by belt drive: P = (F 1 - F 2 ) × V Where, F 1 and F 2 are the tensions on the tight and slack sides of the belt. V = $\frac{\pi D N}{60}$ = velocity of pulley, Where, N = Number of revolutions per minute and D = diameter of the pulley. Calculation: Given: P = 15 kW = 15 × 10 3 W, D = 0.15 m, N = 200 rpm. V=\frac{\pi D N}{60}=\frac{\pi&\times ; 0.15 &\times ; 200}{60}=1.571\;m/s P = (F 1 - F 2 ) × V 15 × 10 3 = (F 1 - F 2 ) × 1.571 (F 1 - F 2 ) = 9549.29 N

Concept: Coefficient of Performance (COP): (COP) Refrigerator = $\frac{Cooling~Effect~{(Q_L)}}{Work~Input~{(W_{Input})}}$ Where, Q L = Heat removed from refrigerator, W = Work input. Calculation: Given: Q L = 300 kJ/min = 5 kJ/sec = 5 kW, Work input = 2 kW (COP) Refrigerator = $\frac{Cooling~Effect~{(Q_L)}}{Work~Input~{(W_{Input})}}=\frac{5}{2}=2.5$

Concept: Thermal efficiency ( η th ) = $\frac{Net~Workdone~{(W_{net})}}{Heat~Added~{(Q_{Add})}}$ We know, Net Work done ( W net ) = Heat Added (Q Add ) - Heat Rejected (Q R ) By Arranging, we get &\eta;_{th}=1-\frac{Q_{Rejected}}{Q_{Added}} Calculation: Given: Q Add = 800 kJ/kg, Q R = 381 kJ/kg. Then, &\eta;_{th}=1-\frac{Q_{Rejected}}{Q_{Added}} &\eta;_{th}=1-\frac{381}{800} η th = 0.52375 ≈ 52.37 %

Explanation: (3i + 1)x + (4i + 4)y + 5 = 0 3ix + x + 4iy + 4y + 5 = 0 Now, we can write this equation as (x + 4y + 5) + i(3x + 4y) = 0 + 0i On comparing LHS and RHS, We have x + 4y + 5 = 0 ---(1) 3x + 4y +0 = 0 ---(2) Subtracting (1) from (2), We get 2x - 5 = 0 ∴ 2x = 5 x = $\frac{5}{2}$ x = 2.5

Conept: f(x, y, z) = x 2 + 5y 2 + 5z 2 - 4x + 40y - 40z + 300 For minimum Value, $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = \frac{\partial f}{\partial z} = 0$ Calculation: Given: $f_x = x^2 - 4x$ ∴ $\frac{\partial f}{\partial x}$ = 2x - 4 = 0 2x - 4 =0 So, x = 2 Similarly, for $f_y = 5y^2 + 40y$ ∴ $\frac{\partial f}{\partial y}$ = 10y + 40 = 0 10y + 40 = 0 So, y = -4 Also, for $f_z = 5z^2 - 40z$ ∴ $\frac{\partial f}{\partial z}$ = 10z - 40 = 0 So, z = 4 Now, Putting values of x, y, z in function f(x, y, z), f(x, y, z) = x 2 + 5y 2 + 5z 2 - 4x + 40y - 40z + 300 f(2, -4, 4) = $2^2 + 5(-4)^2 + 5(4)^2 - 4(2) + 40(-4) - 40(4) + 300$ f(2, -4, 4) = 4 + 80 + 80 - 8 - 160 - 160 + 300 f(2, -4, 4) = 136 ∴ the minimum value of function f defined by f(x, y, z) = 136

Process Control Type-I Error Probability Explained This problem requires us to calculate the combined probability of a Type-I error when multiple independent rules are simultaneously applied to a process control chart. A Type-I error occurs when we incorrectly conclude that a process is out of statistical control when it is actually stable. Understanding Type-I Error in Control Charts In Statistical Process Control (SPC), a Type-I error, often denoted by the Greek letter alpha ($\alpha$), is essentially a false alarm. It means the control chart indicates a process problem (out-of-control state) when no real problem exists. Given Information We have four distinct rules used to detect an out-of-control state. Each rule has its own probability of causing a Type-I error. These probabilities are given as: Rule 1: $P(\text{Type-I Error}_1) = \alpha_1 = 0.005$ Rule 2: $P(\text{Type-I Error}_2) = \alpha_2 = 0.02$ Rule 3: $P(\text{Type-I Error}_3) = \alpha_3 = 0.03$ Rule 4: $P(\text{Type-I Error}_4) = \alpha_4 = 0.05$ Crucially, we are told that these rules are independent of each other. Method for Calculating Combined Probability When multiple rules are used together, the process is considered out-of-control if any one of these rules triggers a signal. Calculating the probability of "at least one" event is often simplified by calculating the probability of the complementary event - that none of the events occur - and subtracting this from 1. In this context, the complementary event is that the process remains in control according to all four rules simultaneously. Let $A_i$ represent the event that Rule $i$ detects an out-of-control state. We are given $P(A_i) = \alpha_i$. The probability that Rule $i$ does not detect an out-of-control state (i.e., it indicates the process is in control) is $P(A_i^c) = 1 - P(A_i) = 1 - \alpha_i$. For Rule 1: $P(A_1^c) = 1 - 0.005 = 0.995$ For Rule 2: $P(A_2^c) = 1 - 0.02 = 0.98$ For Rule 3: $P(A_3^c) = 1 - 0.03 = 0.97$ For Rule 4: $P(A_4^c) = 1 - 0.05 = 0.95$ Since the rules are independent, the probability that the process is considered in control by all rules is the product of these individual probabilities: $P(\text{In Control by All Rules}) = P(A_1^c) \times P(A_2^c) \times P(A_3^c) \times P(A_4^c)$ Step-by-Step Calculation Let's compute the product: $P(\text{In Control by All Rules}) = 0.995 \times 0.98 \times 0.97 \times 0.95$ Performing the multiplication step-by-step: $0.995 \times 0.98 = 0.9751$ $0.9751 \times 0.97 = 0.945847$ $0.945847 \times 0.95 = 0.89855465$ So, the probability that none of the rules signal an out-of-control state is approximately $0.8986$. Final Probability of Overall Type-I Error The overall Type-I error probability, when using all four rules simultaneously, is the probability that at least one rule signals an out-of-control state. This is calculated as: $P(\text{Overall Type-I Error}) = 1 - P(\text{In Control by All Rules})$ $P(\text{Overall Type-I Error}) = 1 - 0.89855465$ $P(\text{Overall Type-I Error}) = 0.10144535$ Rounding this value to three decimal places, we get $0.101$. This represents the total probability of falsely detecting an out-of-control situation when using all four rules concurrently.

Understanding Process Capability Indices This question asks us to calculate two important process capability indices, Cp and Cpk , given specific process parameters and a shift in the process mean. Process capability indices measure a process's ability to produce output within specified limits. Given Process Parameters First, let's identify the information provided: Estimated process standard deviation ($\sigma$): 2 mm Process specification limits: 120 ± 8 mm Shifted process mean ($\mu$): 118 mm From the specification limits, we can determine the Upper Specification Limit (USL) and Lower Specification Limit (LSL): USL = 120 mm + 8 mm = 128 mm LSL = 120 mm - 8 mm = 112 mm The total specification width (SW) is calculated as: $SW = USL - LSL = 128 \, \text{mm} - 112 \, \text{mm} = 16 \, \text{mm}$ Calculating the Cp Index The Cp index measures the potential capability of a process, assuming the process is centered within the specification limits. It compares the specification width to the process width (six \times the standard deviation). The formula for Cp is: $C_p = \frac{USL - LSL}{6 \sigma}$ Substituting the given values: $C_p = \frac{16 \, \text{mm}}{6 \times 2 \, \text{mm}} = \frac{16}{12}$ Simplifying the fraction: $C_p = \frac{4}{3} \approx 1.333$ Calculating the Cpk Index The Cpk index measures the actual capability of a process, taking into account the process mean's position relative to the specification limits. It is the minimum of the capability ratios calculated for the upper and lower specification limits. The formulas for the individual ratios are: Ratio to USL: $\frac{USL - \mu}{3 \sigma}$ Ratio to LSL: $\frac{\mu - LSL}{3 \sigma}$ The formula for Cpk is: $C_{pk} = \min \left( \frac{USL - \mu}{3 \sigma}, \frac{\mu - LSL}{3 \sigma} \right)$ Now, let's calculate these ratios using the given values: Ratio to USL: $\frac{128 \, \text{mm} - 118 \, \text{mm}}{3 \times 2 \, \text{mm}} = \frac{10}{6} = \frac{5}{3} \approx 1.667$ Ratio to LSL: $\frac{118 \, \text{mm} - 112 \, \text{mm}}{3 \times 2 \, \text{mm}} = \frac{6}{6} = 1.000$ Therefore, the Cpk index is the minimum of these two values: $C_{pk} = \min(1.667, 1.000) = 1.000$ Final Result The calculated values for the process capability indices are Cp = 1.333 and Cpk = 1.000. Comparing this to the options provided, the correct option is 1.333, 1.000.

Calculating Components in a Parallel System Using Reliability This problem involves understanding the reliability of a parallel system where multiple identical components work together. If even one component functions, the system functions. We are given the overall system reliability and the reliability of each individual component, and we need to determine how many such components are connected in parallel. Understanding Parallel System Reliability In a parallel system, the system fails only if all individual components fail. The reliability of a single component is often denoted by '$r$', and the reliability of the system with '$n$' identical components in parallel ($R_{system}$) is calculated using the formula: $R_{system} = 1 - (\text{Probability of all components failing})$ Since the components are identical and their failures are assumed to be independent: $\text{Probability of a single component failing} = 1 - r$ Therefore, the probability of all '$n$' components failing is $(1 - r)^n$. Substituting this back into the system reliability formula: $R_{system} = 1 - (1 - r)^n$ Applying the Formula to the Problem We are given: System Reliability ($R_{system}$) = 0.97 Individual Component Reliability ($r$) = 0.68 We need to find the number of components ($n$). Let's plug the values into the formula: $0.97 = 1 - (1 - 0.68)^n$ Step-by-Step Calculation Calculate the probability of a single component failing: $1 - r = 1 - 0.68 = 0.32$ Substitute this value back into the equation: $0.97 = 1 - (0.32)^n$ Rearrange the equation to solve for $(0.32)^n$: $(0.32)^n = 1 - 0.97$ $(0.32)^n = 0.03$ To find '$n$', we use logarithms. Taking the logarithm of both sides (natural log or log base 10 works): $\log((0.32)^n) = \log(0.03)$ Using the logarithm power rule ($\log(a^b) = b \log(a)$): $n \times \log(0.32) = \log(0.03)$ Solve for '$n$': $n = \frac{\log(0.03)}{\log(0.32)}$ Calculate the values of the logarithms: $\log(0.03) \approx -1.5228787$ $\log(0.32) \approx -0.4948500$ Divide the logarithms: $n \approx \frac{-1.5228787}{-0.4948500} \approx 3.077$ Rounding Rule: The question specifies that if the value is a fraction, it should be rounded up to the next higher integer. $n \approx 3.077 \rightarrow \text{Rounded up to } 4$ Final Answer Determination Based on the calculation, the number of identical components required in the parallel system to achieve a reliability of 0.97, when each component has a reliability of 0.68, is approximately 4 after rounding up.

Evaluating Retail Store Site Selection Factors This problem requires selecting the most suitable location for a new retail store from four potential sites: A, B, C, and D. The selection is based on a weighted scoring method. Several key factors are considered, each assigned a specific weight reflecting its importance. Each site is then scored out of 100 for every factor. Data for Site Evaluation The table below outlines the factors, their weights, and the scores awarded to each site: Factor Factor Weight Score for site (out of 100) A B C D Average community income 0.4 60 70 80 50 Demand growth potential 0.13 30 80 50 40 Proximity to existing store 0.35 50 10 40 60 Availability of public transport 0.2 40 30 40 20 Calculating Weighted Scores for Each Site To find the best site, we calculate a weighted score for each location. This involves multiplying the score of each factor by its corresponding weight and summing these products. The formula applied is: $\text{Weighted Score} = \sum_{i=1}^{n} (\text{Score}_i \times \text{Weight}_i)$ Here, $\text{Score}_i$ represents the score for the i-th factor, and $\text{Weight}_i$ is the weight assigned to it. Site A Weighted Score Calculation The weighted score for Site A is calculated as follows: $\text{Score}_A = (60 \times 0.4) + (30 \times 0.13) + (50 \times 0.35) + (40 \times 0.2)$ $\text{Score}_A = 24 + 3.9 + 17.5 + 8 = 53.4$ Site B Weighted Score Calculation The weighted score for Site B is calculated as follows: $\text{Score}_B = (70 \times 0.4) + (80 \times 0.13) + (10 \times 0.35) + (30 \times 0.2)$ $\text{Score}_B = 28 + 10.4 + 3.5 + 6 = 47.9$ Site C Weighted Score Calculation The weighted score for Site C is calculated as follows: $\text{Score}_C = (80 \times 0.4) + (50 \times 0.13) + (40 \times 0.35) + (40 \times 0.2)$ $\text{Score}_C = 32 + 6.5 + 14 + 8 = 60.5$ Site D Weighted Score Calculation The weighted score for Site D is calculated as follows: $\text{Score}_D = (50 \times 0.4) + (40 \times 0.13) + (60 \times 0.35) + (20 \times 0.2)$ $\text{Score}_D = 20 + 5.2 + 21 + 4 = 50.2$ Comparison of Site Scores and Final Selection After calculating the weighted scores for all four sites, we compare them to determine the best location: Site A: 53.4 Site B: 47.9 Site C: 60.5 Site D: 50.2 Site C achieved the highest weighted score ($60.5$). This indicates that, based on the provided factors and their weights, Site C offers the most advantageous profile for opening the new retail store compared to sites A, B, and D.

EOQ Model: Analyzing Cost Impact of Doubled Order Quantity This question explores the effect of an inaccurate order quantity on the total annual cost within the Economic Order Quantity (EOQ) model. Let's break down the EOQ model and calculate the cost impact. Understanding the EOQ Model Components The EOQ model helps determine the optimal order quantity that minimizes the total inventory costs. The primary costs considered are: Ordering Cost (OC): The cost associated with placing an order (e.g., administrative costs, shipping). Holding Cost (HC): The cost of holding inventory (e.g., storage costs, insurance, cost of capital tied up in inventory). Let: D = Annual Demand S = Cost per order H = Holding cost per unit per year Q = Order quantity Calculating Total Annual Cost (C) The total annual cost (excluding the purchase cost of the goods) is the sum of the annual ordering cost and the annual holding cost. The number of orders placed per year is $\frac{D}{Q}$. The average inventory level is $\frac{Q}{2}$. Therefore, the total annual cost formula is: $C = \text{Annual Ordering Cost} + \text{Annual Holding Cost}$ $C = \left( \frac{D}{Q} \right) \times S + \left( \frac{Q}{2} \right) \times H$ Optimal Order Quantity (Q) and Minimum Cost (C) The EOQ formula provides the optimal order quantity $Q^*$ that minimizes the total annual cost $C$. This occurs when the ordering cost equals the holding cost: $\left( \frac{D}{Q^*} \right) \times S = \left( \frac{Q^*}{2} \right) \times H$ The optimal quantity is calculated as: $Q^* = \sqrt{\frac{2DS}{H}}$ At this optimal quantity $Q^*$, the minimum total annual cost $C$ is: $C = \left( \frac{D}{Q^*} \right) \times S + \left( \frac{Q^*}{2} \right) \times H$ Since at the optimum, $\left( \frac{D}{Q^*} \right) × S = \left( \frac{Q^*}{2} \right) × H$, we can write: $C = 2 \times \left( \frac{Q^*}{2} \right) \times H = Q^*H$ Alternatively, $C = 2 × \left( \frac{D}{Q^*} \right) × S$. Substituting $Q^*$, we find $C = \sqrt{2DSH}$. For simplicity \in this problem, we will use the relationships derived from the equality of OC and HC. So, at the optimal level: Ordering Cost = $\frac{C}{2}$ Holding Cost = $\frac{C}{2}$ Calculating the New Cost (C') with Incorrect Quantity (Q') The question states that the order quantity is estimated incorrectly as $Q' = 2Q$, where $Q$ is the optimal quantity ($Q^*$). We need to find the new total annual cost, $C'$, using this incorrect quantity. Using the total annual cost formula with $Q'$: $C' = \left( \frac{D}{Q'} \right) \times S + \left( \frac{Q'}{2} \right) \times H$ Substitute $Q' = 2Q$ into the equation: $C' = \left( \frac{D}{2Q} \right) \times S + \left( \frac{2Q}{2} \right) \times H$ $C' = \frac{1}{2} \left( \frac{D}{Q} \right) \times S + Q \times H$ Now, let's relate this back to the minimum cost $C$. We know that at the optimal quantity $Q$: $\left( \frac{D}{Q} \right) × S = \frac{C}{2}$ $Q × H = C$ (Note: This step is slightly simplified. It should be $(Q/2)H = C/2$, thus $QH = C$). Let's re-evaluate: $C = (D/Q)S + (Q/2)H$. At optimum, $(D/Q)S = (Q/2)H$. So $C = (Q/2)H + (Q/2)H = Q H$. Yes, $QH=C$ is correct. Substitute these values into the expression for $C'$: $C' = \frac{1}{2} \left( \frac{C}{2} \right) + C$ $C' = \frac{C}{4} + C$ $C' = \left( \frac{1}{4} + 1 \right) C$ $C' = \frac{5}{4} C$ $C' = 1.25 C$ Conclusion If the order quantity is estimated incorrectly as $Q' = 2Q$, the corresponding total annual cost $C'$ becomes 1.25 \times the minimum total annual cost $C$. This corresponds to the first option.