The problem states that a box has a length of $(x+5)$ cm, a width of $x$ cm, and a height of 30 cm. The volume of the box is 4500 cm$^3$. We need to find the value of $x$.

1. Problem Description

The problem states that a box has a length of $(x+5)$ cm, a width of $x$ cm, and a height of 30 cm. The volume of the box is 4500 cm$^3$. We need to find the value of $x$.

2. Solution Steps

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

$Volume = length \times width \times height$

We are given:

Length = $(x+5)$ cm

Width = $x$ cm

Height = 30 cm

Volume = 4500 cm$^3$

Plugging these values into the formula:

$4500 = (x+5)(x)(30)$

Divide both sides by 30:

$\frac{4500}{30} = (x+5)(x)$

$150 = x^2 + 5x$

Rearrange the equation into a quadratic equation:

$x^2 + 5x - 150 = 0$

Now we need to solve for $x$. We can factor the quadratic equation:

$(x + 15)(x - 10) = 0$

So, the possible solutions for $x$ are:

$x + 15 = 0 \Rightarrow x = -15$

$x - 10 = 0 \Rightarrow x = 10$

Since the width of the box cannot be negative, we discard the negative solution.

Therefore, $x = 10$.

3. Final Answer

The value of $x$ is 10.