Find the volume of a rectangular prism if the length is [tex]$2x$[/tex], the width is [tex]$4x^2$[/tex], and the height is [tex]$2x^2+x+7$[/tex]. Use the formula [tex]$V=l \cdot w \cdot h$[/tex], where [tex]$l$[/tex] is length, [tex]$w$[/tex] is width, and [tex]$h$[/tex] is height, to find the volume.
A. [tex]$8x^6+8x^4+56x^3$[/tex]
B. [tex]$8x^5+8x^4+56x^3$[/tex]
C. [tex]$16x^6+8x^4+56x^3$[/tex]
D. [tex]$16x^5+8x^4+56x^3$[/tex]
Answer (2)
The volume of the rectangular prism is calculated by multiplying its dimensions, leading to the expression V = 16 x 5 + 8 x 4 + 56 x 3 . The correct multiple-choice option is D. This equation shows how the length, width, and height combine to form the overall volume of the prism.
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Substitute the given length, width, and height into the volume formula: V = ( 2 x ) ( 4 x 2 ) ( 2 x 2 + x + 7 ) .
Multiply the first two terms: V = ( 8 x 3 ) ( 2 x 2 + x + 7 ) .
Distribute the 8 x 3 term: V = 16 x 5 + 8 x 4 + 56 x 3 .
The volume of the rectangular prism is 16 x 5 + 8 x 4 + 56 x 3 .
Explanation
Problem Setup We are given the length l = 2 x , the width w = 4 x 2 , and the height h = 2 x 2 + x + 7 of a rectangular prism. We need to find the volume V using the formula V = lw h .
Substituting Values Substitute the given expressions into the volume formula: V = ( 2 x ) ( 4 x 2 ) ( 2 x 2 + x + 7 )
Multiplying First Terms First, multiply the first two terms: V = ( 8 x 3 ) ( 2 x 2 + x + 7 )
Distributing Now, distribute the 8 x 3 term to each term in the parenthesis: V = 8 x 3 × 2 x 2 + 8 x 3 × x + 8 x 3 × 7
Simplifying Simplify the expression: V = 16 x 5 + 8 x 4 + 56 x 3
Examples
Understanding the volume of rectangular prisms is crucial in various real-world applications. For instance, when designing a storage unit, knowing the length, width, and height allows you to calculate the total storage space available. Similarly, in construction, calculating the volume of materials needed, like concrete for a foundation, ensures accurate ordering and cost estimation. This concept also applies to packaging design, where optimizing the dimensions of a box minimizes material usage while maximizing the space for the product.
