The volume of a cylinder is given by the formula [tex]$V=\pi r^2 h$[/tex], where [tex]$r$[/tex] is the radius of the cylinder and [tex]$h$[/tex] is the height. Suppose a cylindrical can has radius [tex]$(x+8)$[/tex] and height [tex]$(2x+3)$[/tex]. Which expression represents the volume of the can?
[tex]$\pi x^3+19 \pi x^2+112 \pi x+192 \pi$[/tex]
[tex]$2 \pi x^3+35 \pi x^2+80 \pi x+48 \pi$[/tex]
[tex]$2 \pi x^3+35 \pi x^2+176 \pi x+192 \pi$[/tex]
[tex]$4 \pi x^3+44 \pi x^2+105 \pi x+72 \pi$[/tex]
Answer (1)
Substitute the given radius ( x + 8 ) and height ( 2 x + 3 ) into the volume formula V = π r 2 h .
Expand the expression ( x + 8 ) 2 to get x 2 + 16 x + 64 .
Multiply the expanded expression by ( 2 x + 3 ) and simplify to obtain 2 x 3 + 35 x 2 + 176 x + 192 .
Multiply the result by π to get the volume: 2 π x 3 + 35 π x 2 + 176 π x + 192 π .
Explanation
Problem Analysis The volume of a cylinder is given by the formula V = π r 2 h , where r is the radius and h is the height. In this problem, we are given that the radius of the cylindrical can is r = ( x + 8 ) and the height is h = ( 2 x + 3 ) . Our goal is to find an expression that represents the volume of the can.
Volume Calculation To find the volume, we substitute the given expressions for the radius and height into the volume formula: V = π ( x + 8 ) 2 ( 2 x + 3 ) First, we need to expand the expression ( x + 8 ) 2 . Recall that ( a + b ) 2 = a 2 + 2 ab + b 2 . Therefore, ( x + 8 ) 2 = x 2 + 2 ( x ) ( 8 ) + 8 2 = x 2 + 16 x + 64 Now, we substitute this back into the volume formula: V = π ( x 2 + 16 x + 64 ) ( 2 x + 3 ) Next, we multiply the quadratic expression by the binomial expression: V = π [ x 2 ( 2 x + 3 ) + 16 x ( 2 x + 3 ) + 64 ( 2 x + 3 )] V = π [ 2 x 3 + 3 x 2 + 32 x 2 + 48 x + 128 x + 192 ] Combine like terms: V = π [ 2 x 3 + ( 3 x 2 + 32 x 2 ) + ( 48 x + 128 x ) + 192 ] V = π [ 2 x 3 + 35 x 2 + 176 x + 192 ] Finally, distribute the π :
V = 2 π x 3 + 35 π x 2 + 176 π x + 192 π
Final Answer The expression that represents the volume of the can is 2 π x 3 + 35 π x 2 + 176 π x + 192 π .
Examples
Understanding polynomial expressions for volumes can be useful in various real-world scenarios. For example, if you are designing containers of varying sizes where the dimensions are dependent on a variable 'x', knowing the polynomial expression for the volume allows you to quickly calculate the volume for any value of 'x'. This is particularly helpful in manufacturing and packaging industries where optimizing material usage and storage space is crucial. Imagine you're designing cylindrical storage tanks where the radius and height are functions of 'x'. The volume expression helps determine the tank's capacity and cost-effectiveness based on different 'x' values, aiding in making informed design choices.
