Question

The sum of the radii of two concentric circles is 40. A chord with length 36 of the larger circle is trisected by the smaller circle. The area of the smaller circle is k*pi. Find the value of k. Express your answer as an exact decimal.

Please show work. Thank you very much.

Answer 1

r + R = 40
so
r = 40 - R

pi r^2 = k pi
so
k = r^2

right triangles from center to middle of chord
r^2 - 6^2 = R^2 - 18^2
R^2 -r^2 = 324-36 = 288

so we have two equations
R = 40 - r
R^2 -r^2 = 288
then
(40-r)^2 -r^2 = 288
1600 -80 r +r^2-r^2 = 288
80 r = 1312
r = 16.4
r^2 = 268.96
but k = r^2 = 268.96