An island is 2 miles due south of its closest point along a straight shoreline. A visitor is staying in a cabin that is 7 miles west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 5 mph and swims at a rate of 3 mph. How far should the visitor run to minimize the time it takes to reach the island

The visitor needs to run 5.5 miles before he starts swimming to minimize the time it takes to reach the island.

Step-by-step explanation:

The sketch of the setup described is drawn on the attached image to this question.

Let the distance the visitor has to run be x miles

The distance from that endpoint of running to the closest point on the shoreline to the island is then (7 - x) miles

Using Pythagoras theorem, we can then calculate the distance the visitor has to swim

Let that distance be y

y² = 2² + (7 - x)²

y² = 4 + 49 - 14x + x²

y² = x² - 14x + 53

y = √(x² - 14x + 53)

Speed = (distance/time)

Time = (distance/speed)

Running time = (running distance)/(running speed) = (x/5)

Swimming time = (swimming distance)/(swimming speed) = [√(x² - 14x + 53)]/3

Total time to reach the island

= Running time + Swimming time

T = (x/5) + [√(x² - 14x + 53)]/3

we now want to find the distance that the visitor will run to minimize the time to reach the island

At minimum value of T, (dT/dx) = 0

(dT/dx) = (1/5) + [(2x - 14)(x² - 14x + 53)⁻⁰•⁵]/6

multiplying through by 15

15(dT/dx) = 3 + 5[(x - 7)(x² - 14x + 53)⁻⁰•⁵]

So, at minimum value of T, (dT/dx) = 0

0 = 3 + 5[(x - 7)(x² - 14x + 53)⁻⁰•⁵]

-3[(x² - 14x + 53)⁰•⁵] = 5(x - 7)

-3[(x² - 14x + 53)⁰•⁵] = (5x - 35)

Squaring both sides

9(x² - 14x + 53) = (5x - 35)² = 25x² - 350x + 1225

9x² - 126x + 477 = 25x² - 350x + 1225

25x² - 350x + 1225 = 9x² - 126x + 477

16x² - 224x + 748 = 0

Solving this quadratic equation

x = 8.5 or x = 5.5 miles

Since the Total distance from the cabin to the closest shore point to the island is 7 miles, the only feasible answer is x = 5.5 miles.

Therefore, the visitor needs to run 5.5 miles before he starts swimming to minimize the time it takes to reach the island.

Hope this Helps!!!

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