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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!!!
