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). Complete parts (a) through (d) below. a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters/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. T(y) = X b. Verify that the value of y that minimizes the time it takes to retrieve the ball is y = Set the first derivative of T(y) equal to 0 and isolate y2. T'(y) = 0 =0 Find the derivative. ry= Add to both sides and cross-multiply. 12V 2 = Square both sides. y2 = Isolate y2.Solve for y and simplify the result. V = Take the positive square root of both sides. Factor the radicand. Remove a factor of s from each factor in the radicand, then rewrite the denominator as the product of two radicals. This value of y is of T(y). Because T"(y) = ]is for all values of y, it follows that the value of y found in the previous step corresponds to a local,minimum of T(y). c. If the dog runs at 9 m/s and swims at 2 m/s, what ratio - produces the fastest retrieving time? The ratio - ~ produces the fastest retrieving time. (Round to four decimal places as needed.) d. A dog named Elvis who runs at 5.9 m/s and swims at 0.932 m/s was found to use an average ratio _ of 0.159 to retrieve his ball. Does Elvis appear to know calculus? yIUIL UIL NUDILIVE BYUTILIVUL VI NULLI BILLS. Factor the radicand. Remove alactor of s from each factor in the radicand, then rewrite the denominator as the product of two radicals. This value of y is of T(y). Because T"(y) = is for all values of y, it follows that the value of y found in the previous step corresponds to a local minimum of T(y) c. If the dog runs at 9 m/s and swims at 2 m/s, what ratio y produces the fastest retrieving time? The ratio ~ ~ |produces the fastest retrieving time. (Round to four decimal places as needed.) d. A dog named Elvis who runs at 5.9 m/s and swims at 0.932 m/s was found to use an average ratio - of 0.159 to retrieve his ball. Does Elvis appear to know calculus? For Elvis, the ratio ~ ~ |produces the fastest retrieving time. Because this approximately equal to the average ratio - he was found to use, he to know calculus. (Round to four decimal places as needed.)