2152.22 – Catch a Robber and Maximize Your Reward

You have just caught a bank robber. Congratulations! And you are about to collect a reward. But not so fast! The reward RR will be given to you according to this formula: R=x2−6y2−8.R = \dfrac{x^2 - 6}{y^2 - 8}. And the allowed values for xx and yy are:x=−7,−2,0,10,x = -7, -2, 0, 10, and y=−3,−2,1,10.y = -3, -2, 1, 10. Choose xx and yy to maximize your reward.

Solution

We want a big numerator and a small denominator. So we make x2−6x^2 - 6 big and y2−8y^2 - 8 small.

The value x=10x = 10 maximizes the numerator: x2−6=100−6=94.x^2 - 6 = 100 - 6 = 94. The value y=−3y = -3 yields the smallest positivepositive denominator: y2−8=(−3)2−8=1.y^2 - 8 = (-3)^2 - 8 = 1. With these choices, you'll get a reward of 9494.