Question

1. the dimensions of a rectangular prism are $(x + 1)$ feet by $(x + 2)$ feet by 4 feet. the volume of the prism is $(24x - 1)$ cubic feet. what is the value of $x$?

Explanation:

Step1: Recall volume formula

The volume $V$ of a rectangular prism is $V = l \times w \times h$, where $l, w, h$ are its dimensions.
Substitute given values:
$$(x+1)(x+2) \times 4 = 24x - 1$$

Step2: Expand left-hand side

First multiply $(x+1)(x+2)$:
$$(x^2 + 2x + x + 2) \times 4 = 24x - 1$$
Simplify the polynomial inside the parentheses:
$$(x^2 + 3x + 2) \times 4 = 24x - 1$$
Distribute the 4:
$$4x^2 + 12x + 8 = 24x - 1$$

Step3: Rearrange to standard quadratic form

Move all terms to the left-hand side to set the equation to 0:
$$4x^2 + 12x + 8 - 24x + 1 = 0$$
Combine like terms:
$$4x^2 - 12x + 9 = 0$$

Step4: Factor the quadratic

The quadratic is a perfect square trinomial:
$$(2x - 3)^2 = 0$$

Step5: Solve for x

Set the factor equal to 0:
$$2x - 3 = 0$$
Solve for $x$:
$$x = \frac{3}{2}$$

Answer:

$x = \frac{3}{2}$ or $x = 1.5$