The volume of a cube is 2197 cm$^3$. What is its lateral surface area (in cm$^2$)?

Question

Accepted Answer

To determine the lateral surface area of a cube with a given volume, we first need to find the side length of the cube.

1. **Finding the Side Length:**
   - The volume of a cube is given by the formula $ V = s^3 $, where $ s $ is the side length of the cube.
   - We are given that the volume of the cube is 2197 cm$^3$.
   - To find the side length, we take the cube root of the volume: 
     $$
     s = \sqrt[3]{2197}
     $$
   - Calculating the cube root, we find that $ s = 13 $ cm, since $ 13^3 = 2197 $.

2. **Calculating the Lateral Surface Area:**
   - The lateral surface area of a cube is given by the formula $ L = 4s^2 $, where $ s $ is the side length.
   - Substituting the side length we found:
     $$
     L = 4 \times 13^2 = 4 \times 169 = 676 \text{ cm}^2
     $$

3. **Justification for Option 1:**
   - The calculated lateral surface area is 676 cm$^2$, which matches Option 1. Therefore, Option 1 is the correct answer.

4. **Explanation for Incorrect Options:**
   - **Option 2 (576 cm$^2$)**: This value is incorrect because it does not match the calculated lateral surface area of 676 cm$^2$. It could be a result of an incorrect side length calculation.
   - **Option 3 (845 cm$^2$)**: This value is incorrect as it exceeds the calculated lateral surface area. It does not correspond to the correct side length of the cube.
   - **Option 4 (1014 cm$^2$)**: This value is also incorrect as it is significantly higher than the calculated lateral surface area. It suggests an incorrect side length or a misunderstanding of the formula.

In conclusion, the correct lateral surface area of the cube, based on the given volume, is 676 cm$^2$, which corresponds to Option 1.