Let's solve the problem step by step: --- ### **Given:** - **Initial Shaft:** Diameter = 150 mm Length = 500 mm Casting Time = 30 min - **Required Casting:** Diameter = 200 mm Height = 100 mm - **Exponent (n):** n = 2 --- ### **Formula Used** The solidification time for casting is proportional to the square of the volume to surface area ratio: \[ t \propto \left( \frac{V}{A} \right)^n \] So, \[ \frac{t_1}{t_2} = \left(\frac{(V_1/A_1)}{(V_2/A_2)}\right)^n \] Or, \[ t_2 = t_1 \times \left(\frac{V_2/A_2}{V_1/A_1}\right)^n \] --- ### **Calculate for Both Castings** #### **1. Original Shaft:** - Volume: \[ V_1 = \pi \times (75^2) \times 500 = \pi \times 5625 \times 500 = 8,835,750\ \text{mm}^3 \] - Surface Area (Cylinder + 2 ends): \[ A_1 = 2\pi r h + 2\pi r^2 = 2\pi \times 75 \times 500 + 2\pi \times 75^2 \] \[ = 2\pi \times 37,500 + 2\pi \times 5,625 = 75,000\pi + 11,250\pi = 86,250\pi \] \[ A_1 \approx 271,088\ \text{mm}^2 \] \[ \frac{V_1}{A_1} \approx \frac{8,835,750}{271,088} \approx 32.59 \] --- #### **2. Required Casting:** - Volume: \[ V_2 = \pi \times (100^2) \times 100 = \pi \times 10,000 \times 100 = 1,000,000\pi \approx 3,141,593\ \text{mm}^3 \] - Surface Area (Cylinder + 2 ends): \[ A_2 = 2\pi r h + 2\pi r^2 = 2\pi \times 100 \times 100 + 2\pi \times 100^2 \] \[ = 2\pi \times 10,000 + 2\pi \times 10,000 = 20,000\pi + 20,000\pi = 40,000\pi \] \[ A_2 \approx 125,664\ \text{mm}^2 \] \[ \frac{V_2}{A_2} \approx \frac{3,141,593}{125,664} \approx 25.01 \] --- ### **Plug Values into the Formula** \[ t_2 = t_1 \times \left( \frac{V_2/A_2}{V_1/A_1} \right)^n \] \[ t_2 = 30 \times \left( \frac{25.01}{32.59} \right)^2 \] \[ t_2 = 30 \times (.767)^2 \] \[ t_2 = 30 \times .588 \] \[ t_2 \approx 17.64\ \text{minutes} \] --- ### **Check Options** The calculated value does not match any of the provided options directly, but let's double-check the calculations for any possible error in area/volume: #### **Volume Calculations (Double Check):** - For original: \( V_1 = \pi \times 75^2 \times 500 = \pi \times 5,625 \times 500 = \pi \times 2,812,500 \approx 8,835,750 \) - For new: \( V_2 = \pi \times 100^2 \times 100 = \pi \times 10,000 \times 100 = \pi \times 1,000,000 \approx 3,141,593 \) #### **Surface Area Calculations (Double Check):** - For original: \( A_1 = 2\pi \times 75 \times 500 + 2\pi \times 75^2 = 2\pi \times 37,500 + 2\pi \times 5,625 = 75,000\pi + 11,250\pi = 86,250\pi \approx 271,088 \) - For new: \( A_2 = 2\pi \times 100 \times 100 + 2\pi \times 100^2 = 20,000\pi + 20,000\pi = 40,000\pi \approx 125,664 \) #### **Ratios:** - \( (V/A)_1 \approx 32.59 \) - \( (V/A)_2 \approx 25.01 \) #### **Plug in:** - \( \left( \frac{25.01}{32.59} \right)^2 = (.767)^2 = .588 \) - \( t_2 = 30 \times .588 = 17.64 \) --- ### **Re-Examine Options** None of the options matches 17.64 min. Let's check if the exponent n should be 1 instead of 2 (sometimes a typo): If n = 1: \[ t_2 = 30 \times \frac{25.01}{32.59} = 30 \times .767 = 23.01 \] Still, no match. --- ### **Conclusion** **None of the options exactly matches the calculated answer**, so there may be a miscalculation or misinterpretation in the options or in the interpretation of the question. However, based on the formulas and data provided, the closest logical answer should be **17.64 minutes**. If you must choose from the given options, **5.44 minutes** is the only one that makes some sense as being lower than the reference time, but it is much lower than calculated. Please double-check your question for any missing information. **Final Answer: Based on calculation, 17.64 minutes (not listed). Closest option: 5.44 minutes.** --- If you need a step-by-step solution for any clarification, let me know!