To determine which option provides the same volume for both the cone and the cylinder, we need to use the volume formulas for both shapes.
-
Volume of the Cylinder (V_cylinder): \[ V_{cylinder} = \pi r_{cylinder}^2 h_{cylinder} \]
-
Volume of the Cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r_{cone}^2 h_{cone} \]
Given that the volumes of the cone and the cylinder are equal, we set them equal to each other: \[ V_{cylinder} = V_{cone} \] \[ \pi r_{cylinder}^2 h_{cylinder} = \frac{1}{3} \pi r_{cone}^2 h_{cone} \]
Cancelling \(\pi\) from both sides and rearranging gives: \[ 3 r_{cylinder}^2 h_{cylinder} = r_{cone}^2 h_{cone} \]
We will compute this for each of the provided options (assuming we know the cylinder's radius and height):
Assuming some cylinder values, for instance, r_cylinder = 5 and h_cylinder = 12.
-
Option 1: radius of the cone = 5; height of the cone = 4 \[ 3(5^2)(12) = r_{cone}^2(4) \] \[ 3(25)(12) = 5^2(4) \implies 900 = 100 \quad \text{(FALSE)} \]
-
Option 2: radius of the cone = 15; height of the cone = 12 \[ 3(5^2)(12) = (15^2)(12) \] \[ 900 = 225 \cdot 12 \implies 900 = 2700 \quad \text{(FALSE)} \]
-
Option 3: radius of the cone = 15; height of the cone = 4 \[ 3(5^2)(12) = (15^2)(4) \] \[ 900 = 225 \cdot 4 \implies 900 = 900 \quad \text{(TRUE)} \]
-
Option 4: radius of the cone = 5; height of the cone = 12 \[ 3(5^2)(12) = (5^2)(12) \] \[ 900 = 300 \quad \text{(FALSE)} \]
Based on the calculations, the correct response indicating that the radius and height of the cone could be are:
radius of the cone = 15; height of the cone = 4.