A mathematician stands on a beach with his dog at point A. He throws a tennis ba

Question

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 point C is the closest point on the edge of the beach from the tennis ball. a. Assume the dog runs at r m/s and swims at s m/s, where r > s. Also assume the lengths of BC, CD, 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. b. Verify that the value of y that minimizes the time it takes to retrieve the ball is y= (x)/ [((r/s) + 1)^(1/2)] [((r/s)-1)^(1/2)] c. If the dog runs at 8 m/s and swims at 1 m/s, what ratio y/x produces the fastest retrieving time? d. A dog named Elvis who runs at 6.4 m/s and swims at 0.910 m/s was found to use an average ratio y/x of 0.144 to retrieve his ball. Does Elvis appear to know calculus?

Explanation / Answer

distance equals rate times time we will add the rate running times the distance running and the rate swimming times the distance swimming. Distance swimming is the hypotenuse with y and x the two legs. We find it using Pythagorea theorem SQRT(y^2 + x^2) T(y) = r(z - y) + s*SQRT(y^2 + x^2) notice that z and x are constants. As long as z is bigger than y, y does not depend on z. So our time equation is only a function of y, so writing T(y) is correct. As we change y, T will first go done and then go up, where T stops changing is the minium time. dT/dy = -r +(1/2)*s*2y/SQRT(y^2 + x^2) the (1/2) is the starting exponent of the square root. 1/2 - 1 = -1/2, so the derivative has the SQRT in the denominator - this is the -1/2 power. The new 2y is the chain rule working on (y^2 + x^2) r = s*y/SQRT(y^2 + x^2) r/s = y/SQRT(x^2 + y^2) 1 = SQRT(x^2 + y^2)/SQRT(x^2 + y^2) r/s +1 = {y + SQRT(x^2 + y^2)}/SQRT(x^2 + y^2) r/s - 1 = {y - SQRT(x^2 + y^2)}/SQRT(x^2 + y^2) (a+b)(a-b) = a^2 - b^2 (r/s +1)^(1/2) * (r/s -1)^(1/2) = (y^2 - x^2 - y^2)/(x^2 + y^2) = pretty close to x/y sorry, can't quite get it to work out. c) run 8 swim 1 y/x = 1/SQRT(9)SQRT(7) = 0.126 d) does 0.144 = 1/SQRT(6.4/0.910 +1)*SQRT(6.4/0.910 -1) = 0.14364699 yes, very close. The dog is good

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