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
