Question

the lateral surface area of a cylinder is $30\%$ of its surface area. what is the ratio between the base radius and the height of the cylinder?

Answer 1

radius : height = 7 : 3

Explanation:

You want the ratio of the base radius to the height of a cylinder that has 30% of its surface area as lateral area.

Lateral area

The lateral area of a cylinder is ...

LA = 2πrh

Base area

The base area of a cylinder is ...

A = 2(πr²) = 2πr·r

Ratio

Then the ratio of lateral area to total area is ...

`\frac{\text{LA}}{\text{LA+A}}=(2\pi rh)/(2\pi rh+2\pi rr)=(h)/(h+r)=(30)/(100)`

Inverting this relation lets us solve for r/h:

`(h+r)/(h)=(100)/(30)\\\\1+(r)/(h)=1+(70)/(30)\\\\\boxed{(r)/(h)=(7)/(3)}`