Answered

The volume of a cylinder is given by the formula [tex]\( V=\pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius 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?

A. [tex]\( \pi x^3 + 19\pi x^2 + 112\pi x + 192\pi \)[/tex]

B. [tex]\( 2\pi x^3 + 35\pi x^2 + 80\pi x + 48\pi \)[/tex]

C. [tex]\( 2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi \)[/tex]

D. [tex]\( 4\pi x^3 + 44\pi x^2 + 105\pi x + 72\pi \)[/tex]



Answer :

GinnyAnswer
To determine which expression represents the volume of the cylindrical can, let's break down the steps:

1. Identify the given components for radius and height:
- The radius [tex]\( r \)[/tex] is given by [tex]\( r = x + 8 \)[/tex].
- The height [tex]\( h \)[/tex] is given by [tex]\( h = 2x + 3 \)[/tex].

2. Write the formula for the volume [tex]\( V \)[/tex] of a cylinder:
The volume [tex]\( V \)[/tex] is given by:
[tex]\[ V = \pi r^2 h \][/tex]

3. Substitute the given expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ V = \pi (x + 8)^2 (2x + 3) \][/tex]

4. Expand [tex]\( (x + 8)^2 \)[/tex]:
[tex]\[ (x + 8)^2 = x^2 + 16x + 64 \][/tex]

5. Substitute this expansion into the volume formula:
[tex]\[ V = \pi (x^2 + 16x + 64)(2x + 3) \][/tex]

6. Expand the product [tex]\((x^2 + 16x + 64)(2x + 3)\)[/tex]:
[tex]\[ (x^2 + 16x + 64)(2x + 3) = x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3) \][/tex]
[tex]\[ = 2x^3 + 3x^2 + 32x^2 + 48x + 128x + 192 \][/tex]
[tex]\[ = 2x^3 + 35x^2 + 176x + 192 \][/tex]

7. Multiply the expanded polynomial by [tex]\( \pi \)[/tex]:
[tex]\[ V = \pi (2x^3 + 35x^2 + 176x + 192) \][/tex]

8. Compare the expanded volume expression with the given options:
[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]

The expression that matches our expanded result is:
[tex]\[ 2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi \][/tex]

So the correct expression representing the volume of the can is:
[tex]\[ \boxed{2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi} \][/tex]

Other Questions