Question

1. geometry the shower stall has a width of x feet and a depth of (x + 2) feet. the volume of the shower stall is (2x³ + 3x² - 2x) cubic feet.

a. find the height of the shower stall.
b. what is the volume of the shower stall when x = 4?

Explanation:

Step1: Recall the volume formula for a rectangular prism

The volume \(V\) of a rectangular prism is \(V = \text{width}\times\text{depth}\times\text{height}\). Given \(V=2x^{3}+3x^{2}-2x\), \(\text{width}=x\), and \(\text{depth}=x + 2\). Let \(h\) be the height. Then \(h=\frac{V}{\text{width}\times\text{depth}}\). First, factor the volume expression: \(2x^{3}+3x^{2}-2x=x(2x^{2}+3x - 2)\). Factor \(2x^{2}+3x - 2\) further: \(2x^{2}+3x - 2=2x^{2}+4x-x - 2=2x(x + 2)-(x + 2)=(2x - 1)(x + 2)\). So \(V=x(x + 2)(2x - 1)\).

Step2: Calculate the height

Since \(h=\frac{V}{\text{width}\times\text{depth}}\) and \(\text{width}\times\text{depth}=x(x + 2)\), then \(h=\frac{x(x + 2)(2x - 1)}{x(x + 2)}=2x-1\).

Step3: Calculate the volume when \(x = 4\)

Substitute \(x = 4\) into the volume formula \(V=2x^{3}+3x^{2}-2x\). \(V=2\times4^{3}+3\times4^{2}-2\times4\). First, \(4^{3}=64\), \(4^{2}=16\). Then \(V=2\times64+3\times16-8\). \(V = 128+48 - 8\). \(V=168\).

Answer:

a. The height of the shower - stall is \((2x - 1)\) feet.
b. The volume of the shower - stall when \(x = 4\) is \(168\) cubic feet.