Question

The sum of the circumference of two circles is 38π cm and the sum of their areas is 193π cm². Calculate the radii of the circles.​

Answer 1

The radii of the circles are 12 cm and 7cm.

Explanation:

Let the radii of the circles be
`r_(1)\ and\ r_(2)`

Given:

Sum of circumference of two circles= 38π cm

Formula for circumference of a circle with radius r = 2πr

`2\pi r_(1)+2\pi r_(2)=38\pi \\r_(1)+r_(2)=19`------------1

Sum of areas of two circles= 193π cm²

Area of circle with radius r = πr²

`\pi r_(1)^(2)+\pi r_(2)^(2)=193\pi \\r_(1)^(2)+r_(2)^(2)=193`-------2

Substituting r2 = 19 - r1 in equation 2 we get:

`r_(1)^(2)+(19-r_(1))^(2)=193\\2r_(1)^(2)-38r_(1)+168=0\\r_(1)^(2)-19r_(1)+84=0\\(r_(1)-12)(r_(1)-7)=0\\r_(1)=12\ or\ 7`

Then r2 = 19 - r1

`r_(2)=12\ or\ 7`