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

4.11 miles.

Step-by-step 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.

step-by-step explanation:

sas (side angle side)

aa (angle angle)

sss(side side side)

sas find two sides with an angle in between

aa is finding two angles and the third will add up to 180

sss find all sides

step-by-step explanation: