https://file.aiquickdraw.com/mathbot/file/1726809696295p2avczt2.jpg

The problem shows a cylinder with volume formula \( V = \pi r^2 h \), where:

- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder.

From the image, the radius \( r \) is labeled as \( x - 2 \) and the height \( h \) is labeled as \( 3x - 4 \).

To find the volume in terms of \( x \), substitute the expressions for \( r \) and \( h \) into the volume formula:

\[
V = \pi (x - 2)^2 (3x - 4)
\]

Next, expand and simplify the expression:

1. Expand \( (x - 2)^2 \):
\[
(x - 2)^2 = x^2 - 4x + 4
\]

2. Multiply this by \( 3x - 4 \):
\[
V = \pi (x^2 - 4x + 4)(3x - 4)
\]

Now, distribute to simplify further:
\[
V = \pi \left[ (x^2)(3x - 4) + (-4x)(3x - 4) + 4(3x - 4) \right]
\]

Expand each term:
\[
V = \pi \left[ 3x^3 - 4x^2 - 12x^2 + 16x + 12x - 16 \right]
\]

Combine like terms:
\[
V = \pi \left[ 3x^3 - 16x^2 + 28x - 16 \right]
\]

Thus, the volume of the cylinder in terms of \( x \) is:
\[
V = \pi (3x^3 - 16x^2 + 28x - 16)
\]

Would you like a detailed breakdown of any of the steps? Here are a few related questions:

1. How do we find the surface area of the cylinder?
2. What is the volume when \( x = 5 \)?
3. How do you calculate the derivative of the volume with respect to \( x \)?
4. Can you express the volume using different factoring techniques?
5. How does changing the radius or height affect the overall volume?

**Tip:** When dealing with polynomial expressions, always check for common factors that can simplify the calculation.

Find the volume of a cylinder, where the volume is given by V = πr²h, the radius is x - 2, and the height is 3x - 4.

Learn how to calculate the volume of a cylinder using algebraic expressions. Given the radius (x - 2) and height (3x - 4), this detailed solution covers polynomial expansion and simplification. Suitable for Grades 10-12.

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