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Part 1
The volume V of a right circular cylinder of height h and radius r is Upper V equals pi r squared h.  If the height is three times the​ radius, express the volume V as a function of r.
Question content area bottom
Part 1
Upper V left parenthesis r right parenthesisequals
  
enter your response here  ​(Type an exact​ answer, using pi as​ needed.)

The formula for the volume \( V \) of a right circular cylinder is:

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

We are told that the height \( h \) is three times the radius \( r \). This gives us the relationship:

\[
h = 3r
\]

Substituting \( h = 3r \) into the volume formula:

\[
V = \pi r^2 (3r) = 3\pi r^3
\]

Thus, the volume \( V \) as a function of \( r \) is:

\[
V(r) = 3\pi r^3
\]

Let me know if you'd like any further explanation!

Here are some related questions to explore:

1. What is the volume of the cylinder when \( r = 2 \) units?
2. How does the volume change if the height is doubled?
3. Can you express the surface area of the same cylinder as a function of \( r \)?
4. What happens to the volume if the radius is halved?
5. How does the volume formula differ for a cone?

**Tip:** When the height is proportional to the radius, simplifying the volume formula early helps save time!

The volume V of a right circular cylinder of height h and radius r is V = πr^2h. If the height is three times the radius, express the volume V as a function of r.

Explore how to express the volume of a right circular cylinder as a function of its radius, using the relationship between height and radius. This problem involves geometry and algebra, suitable for students in Grades 9-11.

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