Consider two circles. A smaller one and a larger one. If the larger one has a radius that is 8 ft longer than the smaller one, and if the ratio of the circumference is 3:1, then what are the radii of the two circles?

If the smaller radius is r, then

r+8 = 3r

To determine the radii of the two circles, we can set up a system of equations based on the given information.

Let's assume the radius of the smaller circle is represented by 'r'. According to the problem, the radius of the larger circle is 8 ft longer than the smaller circle's radius, so it can be represented as 'r + 8'.

Next, we have the ratio of the circumference of the larger circle to the circumference of the smaller circle, which is given as 3:1. The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius.

Setting up the equation for the larger circle:
Circumference of larger circle = 2π(r + 8)

Setting up the equation for the smaller circle:
Circumference of smaller circle = 2πr

We are given that the ratio of these two circumferences is 3:1:
(2π(r + 8)) / (2πr) = 3/1

By canceling out the common factors, we get:
(r + 8) / r = 3/1

Next, we can cross-multiply to solve for 'r':
1(r + 8) = 3r

Expanding the equation:
r + 8 = 3r

Bringing the 'r' term to one side of the equation:
3r - r = 8

Simplifying the equation:
2r = 8

Finally, solving for 'r':
r = 8/2
r = 4

So, the radius of the smaller circle is 4 ft.

To find the radius of the larger circle, we can substitute this value back into the equation:
r + 8 = 4 + 8 = 12

Therefore, the radius of the larger circle is 12 ft.