AMC 10B/12B 2024 (Problem 14/9)

A dartboard is the region

B

in the coordinate plane consisting of points

(x,y)

such that

|x|+|y|\le 8

. A target

T

is the region where

(x^2+y^2-25)^2\le 49

. A dart is thrown and lands at a random point in

B

. The probability that the dart lands in

T

can be expressed as

\frac{m}{n}\cdot\pi

, where

m

and

n

are relatively prime positive integers. What is

m+n

\text{(A)}\;39

\text{(B)}\;71

\text{(C)}\;73

\text{(D)}\;75

\text{(E)}\;135