Question

a cylinder is shown below.
(image of a cylinder with r = 3 m, h = 6 m)
what are the dimensions of a cone with the same volume?
a cone with the same volume as the cylinder will have a radius of 3 m (dropdown) and a height of (dropdown with options: 1 m, 3 m, 6 m, 9 m, 12 m, 18 m)

Explanation:

Step1: Recall Volume Formulas

Volume of cylinder: \( V_{cylinder} = \pi r^2 h \)
Volume of cone: \( V_{cone} = \frac{1}{3} \pi r^2 h \)

Step2: Calculate Cylinder Volume

Given \( r = 3 \, \text{m} \), \( h = 6 \, \text{m} \):
\( V_{cylinder} = \pi (3)^2 (6) = 54\pi \, \text{m}^3 \)

Step3: Set Cone Volume Equal

Let cone have \( r = 3 \, \text{m} \) (same as cylinder). Let cone height be \( h_{cone} \).
\( \frac{1}{3} \pi (3)^2 h_{cone} = 54\pi \)

Step4: Solve for \( h_{cone} \)

Simplify: \( \frac{1}{3} \pi (9) h_{cone} = 54\pi \)
\( 3\pi h_{cone} = 54\pi \)
Divide both sides by \( 3\pi \): \( h_{cone} = 18 \, \text{m} \)

Answer:

18 m