Question

A cone has twice the base diameter and three-fourths the height of a cylinder. Can you tell what the relationship between the volume of the cone and the volume of the cylinder is? Explain your reasoning.

Answer 1

Answer:

Equal volumes

Step-by-step explanation:

Given

$h \to$ height of cone

$d \to$ diameter of cone

$H \to$ height of cylinder

$D \to$ diameter of cylinder

Such that:

$d = 2D$

$h =\frac{3}{4}H$

Required

The relationship between the volumes

The volume of a cylinder is:

$V_1 = \pi R^2H$

Where

$R = 0.5D$

So:

$V_1 = \pi (0.5D)^2H$

$V_1 = \pi *0.25*D^2H$

$V_1 = 0.25\pi D^2H$

The volume of the cone is:

$V_2 = \frac{1}{3}\pi r^2h$

Where

$r =0.5d$

$r = 0.5 * 2D$

$r = D$

and

$h =\frac{3}{4}H$

So, we have:

$V_2 = \frac{1}{3}\pi * D^2 * \frac{3}{4}H$

$V_2 = \frac{1}{4}\pi * D^2H$

$V_2 = 0.25\pi * D^2H$

$V_2 = 0.25\pi D^2H$

So, we have:

$V_1 = 0.25\pi D^2H$

$V_2 = 0.25\pi * D^2H$

$V_1 = V_2$