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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 find the volume of the cylindrical can, we use the volume formula for a cylinder:

[tex]\[ V = \pi r^2 h \][/tex]

Given:
- Radius [tex]\( r = x + 8 \)[/tex]
- Height [tex]\( h = 2x + 3 \)[/tex]

First, we substitute these into the volume formula:

[tex]\[ V = \pi (x + 8)^2 (2x + 3) \][/tex]

Now, let's expand the expression [tex]\( (x + 8)^2 \)[/tex]:

[tex]\[ (x + 8)^2 = (x + 8)(x + 8) = x^2 + 16x + 64 \][/tex]

Next, multiply this result by [tex]\( 2x + 3 \)[/tex]:

[tex]\[ \pi (x^2 + 16x + 64)(2x + 3) \][/tex]

First, distribute [tex]\( 2x \)[/tex] across [tex]\( x^2 + 16x + 64 \)[/tex]:

[tex]\[ 2x(x^2 + 16x + 64) = 2x^3 + 32x^2 + 128x \][/tex]

Then, distribute [tex]\( 3 \)[/tex] across [tex]\( x^2 + 16x + 64 \)[/tex]:

[tex]\[ 3(x^2 + 16x + 64) = 3x^2 + 48x + 192 \][/tex]

Combine the results by adding like terms:

[tex]\[ 2x^3 + 32x^2 + 128x + 3x^2 + 48x + 192 \][/tex]

Simplify by combining like terms:

[tex]\[ 2x^3 + (32x^2 + 3x^2) + (128x + 48x) + 192 = 2x^3 + 35x^2 + 176x + 192 \][/tex]

Thus, the expression for the volume of the can is:

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

So the expression that represents the volume of the can is:

[tex]\[ 2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi \][/tex]

Therefore, the correct answer is:

[tex]\[ 2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi \][/tex]

From the given options, this matches:
[tex]\[ 2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi \][/tex]

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