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?
$\bigcirc\\ 10a^3 + 22a^2 - 360a - 792$
$\bigcirc\\ 10a^3 + 67a^2 - 90a - 792$
$\bigcirc\\ 10a^3 + 139a^2 + 606a + 792$
$\bigcirc\\ 10a^3 + 91a^2 + 54a - 792$

Explanation:

Step1: Substitute the values into the volume formula

The volume formula for a rectangular prism is \( V = lwh \). We substitute \( l = 2a + 11 \), \( w = 5a - 12 \), and \( h = a + 6 \) into the formula, so we get \( V=(2a + 11)(5a - 12)(a + 6) \).

Step2: Multiply the first two binomials

First, multiply \( (2a + 11) \) and \( (5a - 12) \) using the distributive property (FOIL method):
\[

$$\begin{align*} (2a + 11)(5a - 12)&=2a\times5a+2a\times(-12)+11\times5a + 11\times(-12)\\ &=10a^{2}-24a + 55a-132\\ &=10a^{2}+31a - 132 \end{align*}$$

\]

Step3: Multiply the result by the third binomial

Now, multiply \( (10a^{2}+31a - 132) \) and \( (a + 6) \):
\[

$$\begin{align*} &(10a^{2}+31a - 132)(a + 6)\\ =&10a^{2}\times a+10a^{2}\times6+31a\times a+31a\times6-132\times a-132\times6\\ =&10a^{3}+60a^{2}+31a^{2}+186a-132a - 792\\ =&10a^{3}+(60a^{2}+31a^{2})+(186a - 132a)-792\\ =&10a^{3}+91a^{2}+54a - 792 \end{align*}$$

\]

Wait, there is a mistake in the above calculation. Let's recalculate the multiplication of \( (2a + 11)(5a - 12) \) correctly:

\[

$$\begin{align*} (2a + 11)(5a - 12)&=2a\times5a+2a\times(-12)+11\times5a+11\times(-12)\\ &=10a^{2}-24a + 55a-132\\ &=10a^{2}+31a - 132 \end{align*}$$

\]

Then multiply \( (10a^{2}+31a - 132) \) by \( (a + 6) \):

\[

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

\]

Wait, but let's check again. Wait, maybe I made a mistake in the first multiplication. Let's do \( (2a + 11)(5a - 12) \) again:

\( 2a\times5a = 10a^{2} \), \( 2a\times(-12)=-24a \), \( 11\times5a = 55a \), \( 11\times(-12)=-132 \). Then \( -24a+55a = 31a \), so \( (2a + 11)(5a - 12)=10a^{2}+31a - 132 \) is correct. Then multiplying by \( (a + 6) \):

\( 10a^{2}\times a=10a^{3} \), \( 10a^{2}\times6 = 60a^{2} \), \( 31a\times a=31a^{2} \), \( 31a\times6 = 186a \), \( -132\times a=-132a \), \( -132\times6=-792 \). Then combine like terms: \( 60a^{2}+31a^{2}=91a^{2} \), \( 186a-132a = 54a \). So the result is \( 10a^{3}+91a^{2}+54a - 792 \). Wait, but let's check the options. The fourth option is \( 10a^{3}+91a^{2}+54a - 792 \). Wait, but let's check the multiplication again. Wait, maybe I made a mistake in the sign when multiplying \( (2a + 11)(5a - 12) \). Wait, no, \( 11\times(-12) = -132 \) is correct. Then \( 2a\times(-12)=-24a \), \( 11\times5a = 55a \), so \( -24a + 55a = 31a \). Then multiplying by \( (a + 6) \):

\( (10a^{2}+31a - 132)(a + 6)=10a^{2}(a + 6)+31a(a + 6)-132(a + 6)=10a^{3}+60a^{2}+31a^{2}+186a-132a - 792=10a^{3}+91a^{2}+54a - 792 \). So the correct expression is the fourth option.

Wait, but let's check with another approach. Let's multiply \( (5a - 12)(a + 6) \) first, then multiply by \( (2a + 11) \).

Step2 (Alternative): Multiply \( (5a - 12) \) and \( (a + 6) \)

\[

$$\begin{align*} (5a - 12)(a + 6)&=5a\times a+5a\times6-12\times a-12\times6\\ &=5a^{2}+30a-12a - 72\\ &=5a^{2}+18a - 72 \end{align*}$$

\]

Step3 (Alternative): Multiply the result by \( (2a + 11) \)

Now multiply \( (2a + 11) \) and \( (5a^{2}+18a - 72) \):
\[

$$\begin{align*} &(2a + 11)(5a^{2}+18a - 72)\\ =&2a\times5a^{2}+2a\times18a+2a\times(-72)+11\times5a^{2}+11\times18a+11\times(-72)\\ =&10a^{3}+36a^{2}-144a+55a^{2}+198a - 792\\ =&10a^{3}+(36a^{2}+55a^{2})+(-144a + 198a)-792\\ =&10a^{3}+91a^{2}+54a - 792 \end{align*}$$

\]

Yes, this confirms the result. So the volume is \( 10a^{3}+91a^{2}+54a - 792 \).

Answer:

The expression that represents the volume of the box is \( \boldsymbol{10a^{3}+91a^{2}+54a - 792} \) (the fourth option: 10\(a^3\)+91\(a^2\)+54\(a\)−792).