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The volume of a cylinder is given by the formula , where *r* is the radius of the cylinder and *h* is the height. Suppose a cylindrical can has radius (*x* + 8) and height (2*x* + 3). Which expression represents the volume of the can?


The volume \( V \) of a cylinder is given by the formula:

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

In this case, the radius \( r \) is \( x + 8 \) and the height \( h \) is \( 2x + 3 \).

To find the expression for the volume of the can, substitute the given values of \( r \) and \( h \) into the formula:

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

Now, let's expand this expression step by step.

1. Expand \( (x + 8)^2 \):

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

2. Now, multiply the result by \( (2x + 3) \):

\[
(x^2 + 16x + 64)(2x + 3) = x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3)
\]

3. Expanding each term:

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

4. Combine like terms:

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

So, the expression for the volume \( V \) of the cylindrical can is:

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

This is the expanded form of the volume of the cylinder.

Do you have any further questions or need more details?

Here are five related questions:
1. How does changing the radius affect the volume of the cylinder?
2. What happens to the volume if both the radius and height are doubled?
3. Can you express the volume formula in terms of the diameter instead of the radius?
4. How can you calculate the surface area of the cylinder using the given dimensions?
5. What is the volume if \( x = 1 \)?

**Tip:** When dealing with polynomials, always expand and combine like terms to simplify your expressions.

Learn how to calculate the volume of a cylindrical can with a given radius and height using the formula V = π (x + 8)^2 (2x + 3). This problem involves geometry and polynomials, suitable for high school students.

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