A small island is 2 km off shore in a large lake. A woman on the island can row her boat 10 km/h and can run at a speed of 20 km/h. If she rows to the closest point of the straight shore, she will land 6 km from a village on the shore.
How far from the village should she land to reach it most quickly by a combination of rowing and running?
work this one just like the last. But time = distance/speed, rather than cost=distance*rate.
To find the distance from the village where the woman should land to reach it most quickly, we need to determine the optimal point along the shoreline for her to transition from rowing to running.
Let's assume the distance from the village where she should land is x km.
The time it takes for her to row from the island to the closest point on the shoreline is given by the equation:
Time taken for rowing = Distance / Speed = 2 / 10 = 0.2 hour
Once she reaches the shore, she needs to run the remaining distance to reach the village. The running time is given by the equation:
Time taken for running = Distance / Speed = (6 - x) / 20
The total time taken for travel is the sum of the rowing time and the running time:
Total time = Time taken for rowing + Time taken for running = 0.2 + (6 - x) / 20
To find the point where the total time is minimized, we can find the derivative of the total time equation with respect to x and set it equal to zero, then solve for x:
d(Total time) / dx = d(0.2 + (6 - x) / 20) / dx = -1/20 = 0
Simplifying the equation, we get:
-1/20 = 0
This equation has no solution, which means the total time equation has no minimum or maximum value. Since the woman wants to reach the village as quickly as possible, she should row directly towards the village. Thus, the distance from the village where she should land is 6 km.
Therefore, in order to reach the village most quickly, the woman should row straight to the point 6 km from the village on the shoreline.