The volume of a rectangular prism is given by the formula [tex]$V=l w h$[/tex], where [tex]$l$[/tex] is the length of the prism, [tex]$w$[/tex] is the width, and [tex]$h$[/tex] is the height. Suppose a box in the shape of a rectangular prism has length ([tex]$2 a+11$[/tex]), width ([tex]$5 a-12$[/tex]), and height ([tex]$a+6$[/tex]). Which expression represents the volume of the box?
A. [tex]$10 a^3+22 a^2-360 a-792$[/tex]
B. [tex]$10 a^3+67 a^2-90 a-792$[/tex]
C. [tex]$10 a^3+139 a^2+606 a+792$[/tex]
D. [tex]$10 a^3+91 a^2+54 a-792$[/tex]
Answer (1)
Substitute the given expressions for length, width, and height into the volume formula: V = ( 2 a + 11 ) ( 5 a − 12 ) ( a + 6 ) .
Expand the expression ( 2 a + 11 ) ( 5 a − 12 ) to get 10 a 2 + 31 a − 132 .
Multiply the result by ( a + 6 ) to get 10 a 3 + 91 a 2 + 54 a − 792 .
The expression representing the volume of the box is 10 a 3 + 91 a 2 + 54 a − 792 .
Explanation
Understanding the Problem We are given the volume of a rectangular prism as V = lw h , where l is the length, w is the width, and h is the height. We are given the length l = ( 2 a + 11 ) , the width w = ( 5 a − 12 ) , and the height h = ( a + 6 ) . Our goal is to find the expression that represents the volume of the box.
Substituting the Expressions We need to substitute the given expressions for length, width, and height into the volume formula: V = ( 2 a + 11 ) ( 5 a − 12 ) ( a + 6 ) .
Expanding the Expression - Part 1 Now, we need to expand the expression. First, let's multiply ( 2 a + 11 ) ( 5 a − 12 ) : ( 2 a + 11 ) ( 5 a − 12 ) = 2 a ( 5 a ) + 2 a ( − 12 ) + 11 ( 5 a ) + 11 ( − 12 ) = 10 a 2 − 24 a + 55 a − 132 = 10 a 2 + 31 a − 132.
Expanding the Expression - Part 2 Next, we multiply the result by ( a + 6 ) : ( 10 a 2 + 31 a − 132 ) ( a + 6 ) = 10 a 2 ( a ) + 10 a 2 ( 6 ) + 31 a ( a ) + 31 a ( 6 ) − 132 ( a ) − 132 ( 6 ) = 10 a 3 + 60 a 2 + 31 a 2 + 186 a − 132 a − 792 = 10 a 3 + 91 a 2 + 54 a − 792.
Final Expression Therefore, the expression that represents the volume of the box is 10 a 3 + 91 a 2 + 54 a − 792 .
Examples
Understanding polynomial expressions for volumes is crucial in various real-world applications. For instance, when designing packaging, engineers use these formulas to optimize the dimensions of boxes to minimize material usage while maintaining a specific volume. Similarly, in architecture, calculating the volume of rooms or buildings helps in determining heating and cooling requirements, ensuring energy efficiency. In logistics, knowing the volume of cargo containers allows for efficient space utilization in ships, trains, and trucks, reducing transportation costs.
