Question

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

Answer 1

Answer:

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!!!