Brandon is on one side of a river that is 50 m wide and wants to reach a point 200 m downstream on the opposite side as quickly
as possible by swimming diagonally across the river and then running the rest of the way. Find the minimum amount of time if Brandon can swim at 1.5 m/s and run at 5 m/s. (Round your answer to two decimal places.)
2 answers:
Answer:
71.8 seconds
Step-by-step explanation:
In the diagram, the plot if the situation is shown. Brandon wants to go from A to C swimming at 1.5 m/s and from C to D running at 5 m/s.
From speed definition:
time = distance/speed
From pythagorean theorem
AC = √(x² + 50²)
Then the distance AC is done in:
time = √(x² + 50²)/1.5 (in seconds)
On the other hand, the distance CD is covered in:
time = (200 - x)/5 (in seconds)
The total time is
f(x) = √(x² + 50²)/1.5 + (200 - x)/5
We want to optimize it, then we need to find its first derivative and equalize it to zero:
f(x) = √(x² + 50²)/1.5 + (200 - x)/5
f(x) = √(x² + 50²)/1.5 + 40 - x/5
f'(x) = x/[1.5*√(x² + 50²)] - 1/5 = 0
x/[1.5*√(x² + 50²)] = 1/5
5*x = 1.5*√(x² + 50²)
5²*x² = 1.5²*(x² + 50²)
25*x² - 2.25*x² = 1.5²*50²
22.75*x² = 5625
x = √(5625/22.75)
x = 15.72
(the negative result is not taking into account because that solution of the square root doesn't have physical sense for the problem)
Then the minimum amount of time is:
f(15.72) = √(15.72² + 50²)/1.5 + (200 - 15.72)/5 = 71.8 seconds
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