Question

a right pyramid has a rectangular base with a length that is 3 times its width. the height is 4 centimeters greater than the width of the base. let ( x = ) the width of the base. which formula can be used to determine the volume, ( v ), of the pyramid? ( v = 3x^{3}+12x^{2} ) ( v = 3x^{3}-12x^{2} ) ( v = x^{3}-4x^{2} ) ( v = x^{3}+4x^{2} )

Explanation:

Step1: Find the length and height of the pyramid

Given \(x\) is the width of the base.
The length of the base \(l = 3x\) (since length is 3 times the width).
The height of the pyramid \(h=x + 4\) (since height is 4 centimeters greater than the width).

Step2: Recall the volume formula for a pyramid

The volume formula for a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the area of the base.
For a rectangular base, \(B = l\times w\). Substituting \(l = 3x\) and \(w=x\), we get \(B=3x\times x=3x^{2}\).

Step3: Substitute \(B\) and \(h\) into the volume formula

Substitute \(B = 3x^{2}\) and \(h=x + 4\) into \(V=\frac{1}{3}Bh\).

$$
LATEXBLOCK0
$$

Answer:

\(V=x^{3}+4x^{2}\) (the fourth option)