Question

The area of a circle is directly

proportional to the square of its
radius. A circle with a radius of 2
cm has an area of 12.566 cm^2.
What is the radius of a circle with
an area of 78.54 cm^2?
[?] cm
Round to the nearest whole number.

Answer 1

`radius=5\ cm`

Explanation:

Let r be the radius of the circle and A be the area of the circle.

Given:

`r_(1) = 2\ cm, A_(1)=12.566\ cm^(2)`

And
`r_(2) = ?, A_(2)=78.54\ cm^(2)`

The area of a circle is directly proportional to the square of its radius.

A ∝
`r^(2)`

`A = kr^(2)`-----------(1)

Where k is the constant of proportionality

Find constant value by substituting
`r_(1) = 2\ cm, A_(1)=12.566\ cm^(2)`

in equation 1.

`A = kr^(2)`

`A_(1)=k(r_(1))^(2)`

`12.566=k* 2^(2)`

`12.566=k* 4`

`k=(12.566)/(4)`

`k=3.141`

Find
`r_(2)` by substituting k and
`A_(2)` value in equation 1.

`A = kr^(2)`

`A_(2)=k(r_(2))^(2)`

`12.566=3.141* (r_(2))^(2)`

`(r_(2))^(2)=(78.54)/(3.141)`

`(r_(2))^(2)=25.004`

where: 25.004 ≅ 25

`(r_(2))^(2)=25`

`r_(2)=√(25)`

`r_(2)=5\ cm`

Therefore; the radius of the circle is 5 cm.