Question

using $v = lwh$, what is an expression for the volume of the following prism? image of a rectangular prism with labels: length $\frac{d - 2}{3d - 9}$, width $\frac{4}{d - 4}$, height $\frac{2d - 6}{2d - 4}$ multiple choice options: 1. $\frac{4(d - 2)}{3(d - 3)(d - 4)}$ 2. $\frac{4d - 8}{3(d - 4)^2}$ 3. $\frac{4}{3d - 12}$ 4. $\frac{1}{3d - 3}$

Explanation:

Step1: Identify length, width, height

Let \( l=\frac{d - 2}{3d - 9} \), \( w=\frac{4}{d - 4} \), \( h=\frac{2d - 6}{2d - 4} \).

Step2: Factor denominators/numerators

Factor: \( 3d - 9 = 3(d - 3) \), \( 2d - 6 = 2(d - 3) \), \( 2d - 4 = 2(d - 2) \).
So \( l=\frac{d - 2}{3(d - 3)} \), \( h=\frac{2(d - 3)}{2(d - 2)}=\frac{d - 3}{d - 2} \).

Step3: Apply volume formula \( V = lwh \)

Substitute into \( V=lwh \):
\( V=\frac{d - 2}{3(d - 3)}\times\frac{4}{d - 4}\times\frac{d - 3}{d - 2} \).

Step4: Simplify the expression

Cancel \( (d - 2) \) and \( (d - 3) \):
\( V=\frac{4}{3(d - 4)}=\frac{4}{3d - 12} \).

Answer:

\(\frac{4}{3d - 12}\) (corresponding to the option \(\boldsymbol{\frac{4}{3d - 12}}\))