A cone has a volume of V cubic units. Another cone has the same height and 1/3 of the radius of the original cone. Write an expression for its volume.

Steps to Solve

1. Write the formula for the volume of a cone

The volume $V$ of a cone is given by the formula $V = \frac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height.

2. Define the volume of the first cone

Let $V_1$ be the volume of the first cone with radius $r_1$ and height $h_1$. So, $V_1 = \frac{1}{3}\pi r_1^2 h_1$. We are given that $V_1 = V$.

3. Define the volume of the second cone

Let $V_2$ be the volume of the second cone with radius $r_2$ and height $h_2$. So, $V_2 = \frac{1}{3}\pi r_2^2 h_2$.

4. Relate the radius and height of the two cones

We are given that the second cone has the same height as the first cone, so $h_2 = h_1$. We are also given that the radius of the second cone is $\frac{1}{3}$ the radius of the first cone, so $r_2 = \frac{1}{3}r_1$.

5. Express the volume of the second cone in terms of the first cone's radius and height

Substitute $r_2 = \frac{1}{3}r_1$ and $h_2 = h_1$ into the formula for $V_2$:

$V_2 = \frac{1}{3}\pi (\frac{1}{3}r_1)^2 h_1 = \frac{1}{3}\pi (\frac{1}{9}r_1^2) h_1 = \frac{1}{27}\pi r_1^2 h_1$

6. Express the volume of the second cone in terms of V

We know that $V = V_1 = \frac{1}{3}\pi r_1^2 h_1$. We want to express $V_2$ in terms of $V$. Notice that $\pi r_1^2 h_1 = 3V$. Substituting this into the equation for $V_2$, we get: $V_2 = \frac{1}{27} \pi r_1^2 h_1 = \frac{1}{27}(3V) = \frac{1}{9}V$