The radius of the larger circle is twice as long as the radius of the smaller circle. The area of the donut is 75 pi . Find the radius of the smaller (inner) circle.?

Question

Noah Atkinson · Accepted Answer

The smaller radius is 5

Let r = the radius of the inner circle. Then radius of the larger circle is #2r# From the reference we obtain the equation for the area of an annulus : #A = pi(R^2-r^2)# Substitute 2r for R: #A= pi( (2r)^2- r^2)# Simplify: #A= pi( (4r^2- r^2)# #A= 3pir^2# Substitute in the given area: #75pi = 3pir^2# Divide both sides by #3pi#: #25 = r^2# #r = 5#

Ella Armstrong · Answer

undefined Let $ R $ be the radius of the larger circle and $ r $ be the radius of the smaller circle.

Given that the area of the donut is $ 75\pi $, we can express this as the difference between the areas of the larger circle and the smaller circle:

$$ \pi R^2 - \pi r^2 = 75\pi $$

Given that the radius of the larger circle is twice as long as the radius of the smaller circle, we can express this as $ R = 2r $.

Substitute $ R = 2r $ into the equation:

$$ \pi (2r)^2 - \pi r^2 = 75\pi $$

Simplify:

$$ 4\pi r^2 - \pi r^2 = 75\pi $$
$$ 3\pi r^2 = 75\pi $$

Divide both sides by $ 3\pi $:

$$ r^2 = 25 $$

Take the square root of both sides:

$$ r = 5 $$

So, the radius of the smaller (inner) circle is $ 5 $.