Question
When the dimensions of a cube are reduced by 4 inches on each side, the surface area of the new cube is 864 square inches. What were the dimensions of the original cube?
Answers
Answered by
GPT-4o mini
To solve for the dimensions of the original cube, let's denote the side length of the original cube as \( x \) inches. When the side length is reduced by 4 inches, the new side length becomes \( x - 4 \) inches.
The surface area \( S \) of a cube with side length \( s \) is given by the formula:
\[
S = 6s^2
\]
For the new cube with side length \( x - 4 \), the surface area is given as:
\[
S = 6(x - 4)^2
\]
We know that this surface area equals 864 square inches:
\[
6(x - 4)^2 = 864
\]
To solve for \( x \), we first divide both sides of the equation by 6:
\[
(x - 4)^2 = 144
\]
Next, we take the square root of both sides:
\[
x - 4 = 12 \quad \text{or} \quad x - 4 = -12
\]
Solving for \( x \) in both cases:
1. For \( x - 4 = 12 \):
\[
x = 12 + 4 = 16
\]
2. For \( x - 4 = -12 \):
\[
x = -12 + 4 = -8
\]
(This solution is not valid since a side length cannot be negative.)
Thus, the only valid solution is:
\[
x = 16
\]
Therefore, the dimensions of the original cube are:
\[
\boxed{16 \text{ inches}}
\]
The surface area \( S \) of a cube with side length \( s \) is given by the formula:
\[
S = 6s^2
\]
For the new cube with side length \( x - 4 \), the surface area is given as:
\[
S = 6(x - 4)^2
\]
We know that this surface area equals 864 square inches:
\[
6(x - 4)^2 = 864
\]
To solve for \( x \), we first divide both sides of the equation by 6:
\[
(x - 4)^2 = 144
\]
Next, we take the square root of both sides:
\[
x - 4 = 12 \quad \text{or} \quad x - 4 = -12
\]
Solving for \( x \) in both cases:
1. For \( x - 4 = 12 \):
\[
x = 12 + 4 = 16
\]
2. For \( x - 4 = -12 \):
\[
x = -12 + 4 = -8
\]
(This solution is not valid since a side length cannot be negative.)
Thus, the only valid solution is:
\[
x = 16
\]
Therefore, the dimensions of the original cube are:
\[
\boxed{16 \text{ inches}}
\]