Question

a cone and a cylinder have the same radius. the height of the cylinder is 2 times the height of the cone. what is the ratio of the volume of the cylinder to the volume of the cone?
a. 2:1
b. 3:1
c. 3:2
d. 6:1

Explanation:

Step1: Recall volume formulas

Volume of cylinder: $V_{cylinder} = \pi r^2 h_{cylinder}$
Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h_{cone}$

Step2: Relate heights

Given $h_{cylinder} = 2h_{cone}$

Step3: Find the ratio

$\frac{V_{cylinder}}{V_{cone}} = \frac{\pi r^2 h_{cylinder}}{\frac{1}{3}\pi r^2 h_{cone}}$
Simplify (cancel $\pi r^2$): $\frac{h_{cylinder}}{\frac{1}{3}h_{cone}}$
Substitute $h_{cylinder} = 2h_{cone}$: $\frac{2h_{cone}}{\frac{1}{3}h_{cone}}$
Cancel $h_{cone}$: $\frac{2}{\frac{1}{3}} = 2\times3 = 6$? Wait, no—wait, wait, mistake here. Wait, no: Wait, $h_{cylinder} = 2h_{cone}$, so substitute:

$\frac{V_{cylinder}}{V_{cone}} = \frac{\pi r^2 (2h_{cone})}{\frac{1}{3}\pi r^2 h_{cone}} = \frac{2}{\frac{1}{3}} = 6$? No, wait, no—wait, the cone's volume is (1/3)πr²h, cylinder is πr²H, where H = 2h. So ratio is (πr²*2h) / ( (1/3)πr²h ) = (2) / (1/3) = 6? But that's not matching options. Wait, no, wait the options have D as 6:1? Wait, let's check again.

Wait, cone volume: (1/3)πr²h_cone
Cylinder volume: πr²h_cylinder = πr²(2h_cone)
Ratio: (πr²2h_cone) / ( (1/3)πr²h_cone ) = (2) / (1/3) = 6. So 6:1? But option D is 6:1. Wait, but maybe I made a mistake? Wait, no—wait, the problem says "the height of the cylinder is 2 times the height of the cone". So H = 2h. Then cylinder volume is πr²2h, cone is (1/3)πr²h. So ratio is (2h) / ( (1/3)h ) = 6. So 6:1. So answer is D.

Wait, but let me check again. Let's take r = 1, h_cone = 1. Then h_cylinder = 2.

Cylinder volume: π1²2 = 2π
Cone volume: (1/3)π1²1 = π/3
Ratio: 2π / (π/3) = 6. So 6:1. So D is correct.

Answer:

D. 6:1