The volume of a cylinder is given by the formula mc023-1.jpg, where r is the radius of the cylinder and h is the height. Suppose a cylindrical can has radius (x + 8) and height (2x + 3). Which expression represents the volume of the can?


The formula for the volume of a cylinder is given by:

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

Where:
- \( r \) is the radius
- \( h \) is the height

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

### Step-by-step Solution:
1. Plug the expressions for \( r \) and \( h \) into the volume formula:
   \[
   V = \pi (x + 8)^2 (2x + 3)
   \]

2. First, expand \( (x + 8)^2 \):
   \[
   (x + 8)^2 = x^2 + 16x + 64
   \]

3. Now substitute the expanded form back into the volume formula:
   \[
   V = \pi (x^2 + 16x + 64)(2x + 3)
   \]

4. Expand the product \( (x^2 + 16x + 64)(2x + 3) \) using distributive property (FOIL method):
   \[
   (x^2 + 16x + 64)(2x + 3) = x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3)
   \]
   Now, expand each term:
   \[
   x^2(2x + 3) = 2x^3 + 3x^2
   \]
   \[
   16x(2x + 3) = 32x^2 + 48x
   \]
   \[
   64(2x + 3) = 128x + 192
   \]

5. Combine all the terms:
   \[
   2x^3 + 3x^2 + 32x^2 + 48x + 128x + 192 = 2x^3 + 35x^2 + 176x + 192
   \]

6. Therefore, the expression for the volume of the can is:
   \[
   V = \pi (2x^3 + 35x^2 + 176x + 192)
   \]

This is the volume expression for the cylindrical can.

Would you like to go over any of the steps in more detail or have any further questions?

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Here are five related questions:
1. How would the volume expression change if the radius was \( x + 4 \) instead of \( x + 8 \)?
2. What is the volume if \( x = 1 \)?
3. Can you derive the surface area of the same cylinder?
4. How does changing the height to \( 4x + 2 \) affect the volume?
5. What are some real-world applications of cylindrical volume calculations?

**Tip:** Always double-check your expanded expressions to ensure no terms are missed during distribution!

The volume of a cylinder is given by the formula mc023-1.jpg, where r is the radius of the cylinder and h is the height. Suppose a cylindrical can has radius (x + 8) and height (2x + 3). Which expression represents the volume of the can?

Explore how to calculate the volume of a cylindrical can with radius (x + 8) and height (2x + 3). This problem involves algebraic expansion, the distributive property, and polynomial operations. Suitable for students in Grades 9-12.

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