To solve this problem, we're tasked with finding the volume of a rectangular prism given its length, width, and height as algebraic expressions. Let's proceed step-by-step:
1. Identify the given expressions:
- Length [tex]\( l = 2a + 11 \)[/tex]
- Width [tex]\( w = 5a - 12 \)[/tex]
- Height [tex]\( h = a + 6 \)[/tex]
2. Recall the formula for the volume of a rectangular prism:
[tex]\[ V = l \times w \times h \][/tex]
3. Substitute the given expressions for [tex]\( l \)[/tex], [tex]\( w \)[/tex], and [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = (2a + 11) \times (5a - 12) \times (a + 6) \][/tex]
4. Expand the expression step by step:
First, we need to find the product of the first two expressions, [tex]\((2a + 11)(5a - 12)\)[/tex].
[tex]\[
(2a + 11)(5a - 12) = 2a(5a - 12) + 11(5a - 12) \\
= 2a \cdot 5a + 2a \cdot (-12) + 11 \cdot 5a + 11 \cdot (-12) \\
= 10a^2 - 24a + 55a - 132 \\
= 10a^2 + 31a - 132 \][/tex]
5. Multiply the result by [tex]\( (a + 6) \)[/tex]:
[tex]\[
(10a^2 + 31a - 132)(a + 6) = 10a^2(a + 6) + 31a(a + 6) - 132(a + 6) \\
= 10a^2 \cdot a + 10a^2 \cdot 6 + 31a \cdot a + 31a \cdot 6 - 132 \cdot a - 132 \cdot 6 \\
= 10a^3 + 60a^2 + 31a^2 + 186a - 132a - 792 \\
= 10a^3 + (60a^2 + 31a^2) + (186a - 132a) - 792 \\
= 10a^3 + 91a^2 + 54a - 792 \][/tex]
6. Simplify the expression to get the final volume:
[tex]\[
V = 10a^3 + 91a^2 + 54a - 792
\][/tex]
So, the expression that represents the volume of the box is:
[tex]\[ 10a^3 + 91a^2 + 54a - 792 \][/tex]
From the given options, the correct choice is:
[tex]\[ \boxed{10a^3 + 91a^2 + 54a - 792} \][/tex]
