The problem asks to find the value of $x$ given that the volume of a box is 252 cubic feet and the dimensions of the box are $7$, $x-3$, and $x-3$ feet.

1. Problem Description

The problem asks to find the value of $x$ given that the volume of a box is 252 cubic feet and the dimensions of the box are $7$, $x-3$, and $x-3$ feet.

2. Solution Steps

The volume of a box is given by the formula:

$Volume = Length \times Width \times Height$

In this problem, we are given the volume and the dimensions, so we can write the equation:

$252 = 7 \times (x-3) \times (x-3)$

$252 = 7(x-3)^2$

Divide both sides by 7:

$36 = (x-3)^2$

Take the square root of both sides:

$\sqrt{36} = \sqrt{(x-3)^2}$

$\pm 6 = x - 3$

We have two possible equations:

$6 = x - 3$

$-6 = x - 3$

Solving for x in the first equation:

$x = 6 + 3$

$x = 9$

Solving for x in the second equation:

$x = -6 + 3$

$x = -3$

Since the length of a side must be positive, we have to check if both values are admissible.

If $x = 9$, then $x - 3 = 9 - 3 = 6$.

Since 6 is positive, $x=9$ is a valid solution.

If $x = -3$, then $x - 3 = -3 - 3 = -6$.

Since -6 is negative, $x=-3$ is not a valid solution.

Therefore, the only valid solution is $x = 9$.

3. Final Answer

9