Optimization: A man on an island 16 miles north of a straight shoreline must reach a point 30 miles east of the closet point on the shore to the island. If he can row at a speed of 3 mph and jog at a speed of 5 mph, where should he land on the shore in order to reach his destination as soon as possible?
My sketch shows the following:
M is the man's position 16 m off shore
A is the point on the shore perpendicular to M
B is the point where he wants to go, AB = 30
Let P be the optimum point between A and B, let
AP = x, then PB = 30-x
distance in water = MP
MP = (x^2 + 256)^(1/2)
time spent rowing = (x^2 + 256)^(1/2) /3
time spent jogging = (30-x)/5
T = total time
= (1/3)(x^2 + 256)^(1/2) + (1/5)(30-x)
dT/dx = (1/6)(x^2 + 256)^(-1/2) (2x) - 1/5
= 0 for a min of T
x/(3√(x^2 + 256) = 1/5
3√(x^2+256) = 5x
square both sides
9x^2 + 2304 = 25x^2
16x^2 = 2304
x^2 = 144
x = 12
so he should land at 30-12 or 18 miles west of his destination
To solve this optimization problem, we need to minimize the time it takes for the man to reach his destination. Let's break down the problem into two parts: rowing and jogging.
1. Rowing: The man is on an island 16 miles north of the shoreline, and he needs to reach a point 30 miles east of the closest point on the shore to the island. The speed at which he can row is 3 mph.
The time it takes to row a distance can be calculated using the formula: Time = Distance / Speed.
Let the distance to be rowed be x miles. The time taken to row x miles is therefore given by: Rowing Time = x / 3.
2. Jogging: Once the man reaches the shore, he needs to jog to his final destination, which is 30 miles east of the closest point on the shore to the island. His jogging speed is 5 mph.
The distance he needs to jog is the distance from the landing point on the shore to the final destination, which is 30 miles in this case. The time taken to jog 30 miles at a speed of 5 mph is: Jogging Time = 30 / 5 = 6 hours.
Now, the total time taken to reach the destination is the sum of the rowing time and the jogging time: Total Time = Rowing Time + Jogging Time.
Substituting the rowing time and jogging time equations, we get: Total Time = x / 3 + 6.
To minimize the total time, we need to find the value of x that minimizes the equation.
To do this, we can take the derivative of the equation with respect to x and set it equal to zero to find the critical point. Solving for x will give us the optimal distance to row.
Taking the derivative: d(Total Time) / dx = 1/3.
Setting the derivative equal to zero: 1/3 = 0.
Since the derivative is a constant, there are no critical points.
Since there are no critical points, we need to consider the boundary points. The man must choose a landing point somewhere on the shoreline, so x must be between 0 and 16.
Let's calculate the total time for the two boundary points:
For x = 0, Total Time = 0 / 3 + 6 = 6 hours.
For x = 16, Total Time = 16 / 3 + 6 = 11.33 hours.
As we can see, for x = 0, the total time is 6 hours, while for x = 16, the total time is 11.33 hours.
Therefore, the man should land at the closest point on the shore to the island, which is 16 miles north of the shore. This will minimize the total time and allow him to reach his destination as soon as possible.