The question involves calculating the time required to solidify a cubic box of length 3 cm filled in 5 minutes with sand casting, assuming the same conditions for a spherical object in the same environment. ### Key Points: - Cubic box side length: 3 cm - Time to solidify cubic box: 5 minutes - Goal: Find the time for a spherical object in the same conditions ### Assumptions: - The rate of heat transfer is proportional to the surface area. - The same material, environment, and conditions apply. ### Step-by-step solution: #### 1. Calculate volume and surface area: - Volume of cube \( V_{cube} = a^3 = 3^3 = 27\,cm^3 \) - Surface area of cube \( A_{cube} = 6a^2 = 6 \times 3^2 = 54\,cm^2 \) #### 2. For the sphere: Let the radius of the sphere be \( r \). - Volume of sphere \( V_{sphere} = \frac{4}{3} \pi r^3 \) - Surface area of sphere \( A_{sphere} = 4 \pi r^2 \) Since the objects are in the same environment: \[ \text{Rate of solidification} \propto \frac{\text{Surface Area}}{\text{Volume}} \] #### 3. Calculate the ratio: \[ \frac{A_{cube}/V_{cube}}{A_{sphere}/V_{sphere}} = \frac{54/27}{4 \pi r^2 / \frac{4}{3} \pi r^3} = \frac{2}{(4 \pi r^2) / (\frac{4}{3} \pi r^3)} = \frac{2}{ (4 \pi r^2) \times (\frac{3}{4 \pi r^3})} = \frac{2}{3/r} = \frac{2r}{3} \] The time to solidify is proportional to this ratio: \[ T_{sphere} = T_{cube} \times \frac{A_{sphere}/V_{sphere}}{A_{cube}/V_{cube}} \] Since the proportionality is inverse (more surface area per volume means faster solidification), the relation becomes: \[ T_{sphere} = T_{cube} \times \frac{A_{cube}/V_{cube}}{A_{sphere}/V_{sphere}} = 5 \text{ min} \times \frac{3}{2r} \] But we need \( r \) such that volume of sphere equals volume of cube: \[ \frac{4}{3} \pi r^3 = 27 \] \[ r^3 = \frac{27 \times 3}{4 \pi} = \frac{81}{4 \pi} \] \[ r = \sqrt[3]{\frac{81}{4 \pi}} \approx \sqrt[3]{\frac{81}{12.566}} \approx \sqrt[3]{6.45} \approx 1.87\,cm \] Now, calculate \( T_{sphere} \): \[ T_{sphere} = 5 \times \frac{3}{2 \times 1.87} \approx 5 \times \frac{3}{3.74} \approx 5 \times .80 \approx 4.00\, \text{min} \] ### **Answer:** **Option: 3 min 20 sec** Closest to the calculated value, thus: ## Correct Option: **3 min 20 sec**