Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>




a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. 

Verified step by step guidance

1. Understand the problem setup: The dog runs from point A to point D along the beach and then swims from point D to point B. We need to find the total time T(y) it takes for the dog to reach the ball.
2. Identify the distances: Let AC = z, CD = y, and BC = x. The dog runs the distance AD = z - y and swims the distance DB = sqrt(x^2 + y^2) using the Pythagorean theorem.
3. Express the time taken for each segment: The time to run from A to D is (z - y)/r, and the time to swim from D to B is sqrt(x^2 + y^2)/s.
4. Combine the times to form the total time function: T(y) = (z - y)/r + sqrt(x^2 + y^2)/s.
5. This function T(y) represents the total time it takes for the dog to reach the ball, depending on the choice of point D along the beach.