Question

An island is 2.6 mi due north of its closest point on a straight shoreline. A visitor staying at a cabin on the shore that is 8 mi from that point. The visitor is planning to go from the cabin to the island. Suppose that the visitor runs at a rate of 6 mph and swims at a rate of 1 mph How far should the visitor run before swimming to minimize the time it takes to reach the island? Assume that the distance from the point on the shore closest to the island and the point at which the visitor begins swimming is z. Then the time spent running by the visitor as a function of u is T, hr Assume that the distance from the the island and the point at which the visitor begins swimming is y. Then the time spent swimming by the visitor as a function of y is hr What is the total time spent running and/or swimming by the visitor as a function of x? T(2) = hr How far should the visitor run from the cabin in order to minimize the time it takes for them to reach the island? What is the minimum amount of time it will take for the visitor to reach the island from the cabin? hr NOTE: you may use decimals in your answers to this problem.

Answer 1

Let on the Kisland " x be the distance from the star dosest point striaght to the point shave line 2.6mi where the visitor stope running running form geo (8-x) mi KR.6²7 i. He is smi "shoreline Cabin 8-X Th= Ihr. 6 Ts = 1/2.6²+22 ar J 6.76+x² hr. T(31) = 8-X 6 t 6.76+x2 At 8-X + any critical point T'() = 0 √6.76+22 2 -ţ+ * * (6-76+225 x 22 = 0 * (6.764 x4) Hellel kell 6.76+22. 35x² = 6.76 x = > / » 3622 6 7 7 169 3/2 25 >o 7 x²= 6.76 13 35 5135 Also, T'(x) = for a yo. 25 (a} ) ..at 13 T(al is minimum, 3 135 :. The visitor should run chos mi ~ 7.5605 mi (correct upto The minimum time taken is 4 + 25 kn ~ 3.89697 hr. (comeet upto 2= - 8 5 D