Question

A rectangular prism and a cylinder have the same height. The length of each side of the prism base is equal to the diameter of the cylinder. Which shape has a greater volume? Drag and drop the labels to explain your answer.

Answer 1

The rectangular prism has the greater volume

Explanation:

Given

Prism:

`Length = Width = a`

`Height = h`

Cylinder

`Diameter = a`

`Height = h`

Required

Shape with greater volume

The volume of a rectangular prism is:

`Volume = Length * Base * Height`

So, we have:

`V_1 = a*a*h`

`V_1 = a^2h`

The volume of a cylinder is:

`Volume = \pi r^2h`

Where

`r = (d)/(2)`

`r = (a)/(2)`

So, we have:

`V_2 = \pi *(a/2)^2 h`

`V_2 = \pi *(a^2)/(4) h`

`\pi = 3.14`

So:

`V_2 = 3.14*(a^2)/(4) h`

`V_2 = (3.14)/(4) a^2h`

`V_2 = 0.785 a^2h`

So, we have:

`V_1 = a^2h`

`V_2 = 0.785 a^2h`

Both expressions have the same factor but different coefficient

The expression with greater coefficient has the greater volume

So, the
`rectangular`
`prism \\` has the
`greater`
`volume`

Because:
`1 > 0.785`