Cube Volume Explanation

This problem involves finding the lateral surface area of a cube when its volume is given. To solve this, we first need to determine the length of one side of the cube using the provided volume.

The volume ($V$) of a cube is calculated by cubing its side length ($a$). The formula is:

$V = a^3$

We are given that the volume of the cube is $42,875 \text{ m}^3$.

Finding the Cube Side Length

To find the side length ($a$), we need to take the cube root of the volume:

$a = \sqrt[3]{V}$

Substitute the given volume into the equation:

$a = \sqrt[3]{42,875 \text{ m}^3}$

To find the cube root of 42,875, we look for a number that, when multiplied by itself three times, results in 42,875. Since the number ends in 5, its cube root must also end in 5. Testing $35$:

$35 \times 35 \times 35 = 1225 \times 35 = 42,875$

Thus, the side length of the cube is $a = 35$ meters.

Cube Lateral Surface Area Calculation

The lateral surface area (LSA) of a cube is the total area of its four vertical faces. The formula for the LSA is:

$LSA = 4a^2$

Now, we substitute the side length we found ($a = 35$ m) into this formula:

$LSA = 4 \times (35 \text{ m})^2$

First, calculate the area of one square face ($a^2$):

$a^2 = (35 \text{ m})^2 = 1225 \text{ m}^2$

Next, multiply the area of one face by 4 to find the total lateral surface area:

$LSA = 4 \times 1225 \text{ m}^2$

$LSA = 4900 \text{ m}^2$

Therefore, the lateral surface area of the cube is $4900 \text{ m}^2$.