Final answer:
To find the optimal landing point and minimize travel time, we use optimization techniques from calculus to first express total time relative to the variable distance x, then find the derivative, set it to zero and solve for x. This x-value is then used to determine the shortest travel time.
Step-by-step explanation:
The student's question involves finding the point along a shore where a person should land to minimize travel time from an island to a town, given different speeds for rowing and running. To solve this, we can use the concept of optimization in calculus, specifically involving right triangles and the Pythagorean theorem.
Let's denote the distance from the island to the landing point as x. The distance from the landing point to the town on shore is then (14 - x). The time taken to row to the landing point is the distance divided by the rowing speed, so it's x/5 hours. The time taken to run to the town is (14 - x)/7 hours. The total time for the trip, T(x), is the sum of both times:
T(x) = x/5 + (14 - x)/7
To minimize T(x), we would take the derivative of T(x) with respect to x, set it equal to zero, and solve for x to find the critical point. The critical point where T'(x) = 0 gives the optimal landing point. We can then plug this x value back into T(x) to find the shortest time.
Finally, the person would row straight to that point and then run straight to the town, minimizing the overall travel time.