Question

the volume of a rectangular prism is given by the formula $v = lwh$, where $l$ is the length of the prism, $w$ is the width, and $h$ is the height. suppose a box in the shape of a rectangular prism has length $(2a + 11)$, width $(5a - 12)$, and height $(a + 6)$. which expression represents the volume of the box?$10a^3+22a^2-360a-792$$10a^3+67a^2-90a-792$$10a^3+139a^2+606a+792$$10a^3+91a^2+54a-792$

Explanation:

Step1: Substitute values into volume formula

$V=(2a+11)(5a-12)(a+6)$

Step2: Multiply first two binomials

First, calculate $(2a+11)(5a-12)$:

$$\begin{align*} (2a)(5a)+(2a)(-12)+11(5a)+11(-12)&=10a^2-24a+55a-132\\ &=10a^2+31a-132 \end{align*}$$

Step3: Multiply result by third binomial

Now multiply $(10a^2+31a-132)(a+6)$:

$$\begin{align*} 10a^2(a)+10a^2(6)+31a(a)+31a(6)-132(a)-132(6)&=10a^3+60a^2+31a^2+186a-132a-792\\ &=10a^3+(60a^2+31a^2)+(186a-132a)-792\\ &=10a^3+91a^2+54a-792 \end{align*}$$

Answer:

D. $10a^3+91a^2+54a-792$