Question

Brandon is on one side of a river that is 80 m wide and wants to reach a point 250 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 2 m/s and run at 4 m/s

Answer 1

Now get S and R in terms of angle "x" using trig

\[S = \frac{80}{\cos x} = 80 \sec x\]
\[R = 250 - 80 \tan x\]
This gives amount of time as function of x:
\[T = 40 \sec x + \frac{250}{4} - 20 \tan x\]
minimize by setting derivative equal to 0
\[\frac{dT}{dx} = 40 \sec x \tan x - 20 \sec^2 x = 0\]
solve for x
\[x = \sin^{-1} \frac{1}{2} = \frac{\pi}{6}\]
plug it in to get min Time
\[T_{\min} = 20 \sqrt{3} + \frac{250}{4}\]