Question

A rock formation is 5 miles due north of its closest point along a straight shoreline. A visitor is staying in an inn that is 7 miles east of that point. The visitor is planning to go from the inn to the rock formation. Suppose the visitor runs at a rate of 6 mph and swims at a rate of 3 mph. How far should the visitor run to minimize the time it takes to reach the rock formation

Answer 1

4.11 miles.

Explanation:

In the image that I am going to attach we can see the description of the statement in a graph, now, the right triangle is formed, and we can apply Pythagoras, and the hypotenuse would be equal to:

(5 ^ 2 + x ^ 2) ^ (1/2)

The speed is equal to:

V = d / t, therefore the time is t = d / V, the total time would then be:

total time = (5 ^ 2 + x ^ 2) ^ (1/2) / 3 + (7-x) / 6

we derive with respect to x, and we are left with:

dt / dx = x / (3 * (5 ^ 2 + x ^ 2) ^ (1/2)) - 1/6 = 0

x / (3 * (5 ^ 2 + x ^ 2) ^ (1/2)) = 1/6

6x / (3 = (5 ^ 2 + x ^ 2) ^ (1/2)

(2 * x) ^ 2 = (5 ^ 2 + x ^ 2)

4 * x ^ 2 = 25 + x ^ 2

3 * x ^ 2 = 25

x ^ 2 = 25/3

x = 2.88

now we replace

7 - x = 7 - 2.89 = 4.11

In other words, the answer is 4.11 miles.