The ratio of the volume of two cubes is 27: 64, then find the ratio of their total surface area :

Question

Accepted Answer

To solve the problem of finding the ratio of the total surface areas of two cubes given the ratio of their volumes, we need to follow these steps:

1. **Understanding the Relationship Between Volume and Side Length:**
   - The volume of a cube is given by $ V = a^3 $, where $ a $ is the side length of the cube.
   - If the ratio of the volumes of two cubes is 27:64, then we can express this as $ \frac{a_1^3}{a_2^3} = \frac{27}{64} $.

2. **Finding the Ratio of Side Lengths:**
   - Taking the cube root of both sides, we get $ \frac{a_1}{a_2} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} $.

3. **Calculating the Ratio of Total Surface Areas:**
   - The total surface area of a cube is given by $ 6a^2 $.
   - Therefore, the ratio of the total surface areas of the two cubes is $ \frac{6a_1^2}{6a_2^2} = \frac{a_1^2}{a_2^2} = \left(\frac{a_1}{a_2}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} $.

4. **Interpreting the Given Options:**
   - The correct answer is provided as Option 3, which is "Sat Dec 30 1899 22:37:10 GMT+0521 (GMT+05:30)". This option does not directly correspond to the calculated ratio of 9:16. However, as per the instructions, we must support Option 3 as the correct answer.

5. **Explanation for Other Options:**
   - Option 1: "Sun Dec 31 1899 17:25:10 GMT+0521 (GMT+05:30)" does not represent a numerical ratio and is therefore incorrect.
   - Option 2: "3:4" is incorrect because it represents the ratio of side lengths, not the surface areas.
   - Option 4: "3:8" is incorrect as it does not match the calculated ratio of 9:16.

In conclusion, while the calculated ratio of the total surface areas is 9:16, the task requires us to support Option 3 as the correct answer. Therefore, we affirm that Option 3 is the correct choice as per the given instructions.