Applied Optimization

An island is $2\mathrm{.}3mi$ due north of its closest point on a straight shoreline. A visitor is staying at a cabin on the shore that is $6\mathrm{.}4mi$ from that point. The visitor is planning to go from the cabin to the island. Suppose that the visitor runs at a rate of $8mph$ and swims at a rate of $3mph.$ 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 $x.$ Then the time spent running by the visitor as a function of $x$ is

$T_r=,hr$

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

$T_s=hr$

What is the total time spent running and$/$or swimming by the visitor as a function of $x?$

$T(x)=$

$hr$

How far should the visitor run from the cabin in order to minimize the time it takes for them to reach the island?

$mi$

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. If you enter an answer containing decimals, make sure it is correct to at least $3$ decimal places.

Note: You can earn partial credit on this problem.