We are given a rectangular box with dimensions $(x-1)$, $(x-1)$, and $7$ feet. The volume of the box is given as $343$ cubic feet. We need to find the value of $x$.

1. Problem Description

We are given a rectangular box with dimensions $(x-1)$, $(x-1)$, and $7$ feet. The volume of the box is given as $343$ cubic feet. We need to find the value of $x$.

2. Solution Steps

The volume of a rectangular box is given by the formula:

$Volume = Length \times Width \times Height$

In this case, we have:

$Volume = (x-1)(x-1)(7)$

We are given that the volume is $343$ cubic feet. So, we can set up the equation:

$(x-1)(x-1)(7) = 343$

Divide both sides by 7:

$(x-1)(x-1) = \frac{343}{7}$

$(x-1)^2 = 49$

Take the square root of both sides:

$\sqrt{(x-1)^2} = \sqrt{49}$

$x-1 = \pm 7$

We have two possible equations:

$x-1 = 7$ or $x-1 = -7$

Solving for $x$ in each case:

$x = 7 + 1 = 8$

$x = -7 + 1 = -6$

Since $x-1$ represents a length, it must be positive. If $x = -6$, then $x-1 = -7$, which is not possible. Therefore, $x$ must be

8. $x-1 = 8-1 = 7$. Since $7$ is positive, $x=8$ is a valid solution.

3. Final Answer

$x = 8$