Question

find the volume of the figure.
4in
9 in
6 in
volume = \square in³
desmos calculator

Explanation:

Step1: Find the radius of the cylinder and cone

The diameter of the base of the cylinder (and cone, since they share the same base) is 4 in, so the radius \( r = \frac{4}{2} = 2 \) in.

Step2: Find the height of the cone

The total height of the figure is 9 in, and the height of the cylinder is 6 in, so the height of the cone \( h_{cone} = 9 - 6 = 3 \) in.

Step3: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \). Substituting \( r = 2 \) in and \( h = 6 \) in, we get \( V_{cylinder} = \pi \times 2^2 \times 6 = 24\pi \) \( \text{in}^3 \).

Step4: Calculate the volume of the cone

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \). Substituting \( r = 2 \) in and \( h = 3 \) in, we get \( V_{cone} = \frac{1}{3}\pi \times 2^2 \times 3 = 4\pi \) \( \text{in}^3 \).

Step5: Calculate the total volume

Add the volume of the cylinder and the volume of the cone: \( V_{total} = V_{cylinder} + V_{cone} = 24\pi + 4\pi = 28\pi \approx 28\times3.14 = 87.92 \) \( \text{in}^3 \).

Answer:

\( 28\pi \) (or approximately \( 87.92 \))