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: : When a cone and cylinder have the same height and radius the cone will fit inside the cylinder. The volume of the cone will be one-third that of the cylinder. If the radius or height are different, then there is no relationship between them.

Explanation:

Answer 2

Equal volumes

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 =(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 = (1)/(3)\pi r^2h`

Where

`r =0.5d`

`r = 0.5 * 2D`

`r = D`

and

`h =(3)/(4)H`

So, we have:

`V_2 = (1)/(3)\pi * D^2 * (3)/(4)H`

`V_2 = (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`