Applied Optimization

Question

A boat on the the ocean is 9 km from the nearest point on a straight shoreline; that point is 2 km from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. If she walks at 5 km/h and rows at 4 km/h. How far will the point on shore be from the restaurant if she plans to minimize her total travel time

Answer 1

Just a rough guess, but probably one mile from the restaurant

time = sqr(81+1)/4 + 1/5 = 2.264 + 2.5 = 2.764 hours total

distance = speed times time, d=rt

sqr(81+x^2)/4 + (2-x)/5 = time

to minimize, take the derivative and set equal to zero and solve for x

multiply by 20 to remove some fractions

20t' = 5sqr(81+x^2)+8-4x = 0

5sqr(81+x^2) = 4x

square both sides

25(81+x^2) = 16x^2

25x^2-16x^2 = 25(81)

9x^2 = 25(81)

x^2 = 25(9)

x =5x3=15

divide by 20 to get 15/20 = .75 miles =

2-.75 = 1 1/4 mile from the restaurant

square root of 81+.75 divided by 4 = 2.2604 hours

1.25/5 = 0.25 hours

for a total 2.5104 hours

which is faster than 1 mile from the restaurant

answer is 1.25 miles from the restaurant to minimize time