Question

A small resort is situated on an island that lies exactly 3 miles from P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest surfing school. A surfer can walk 5 miles per hour along the shore with the surfboard and can paddle 3 miles per hour to the island. The surfer can walk partway from the school along the shore and then paddle to the resort. How far down the shoreline from P should the surfer start surfing to get to the island as quick as possible. x: distance in miles from P, from where the surfer should start surfing . Surfy, Boogie, Windy, Sunny and Wavy individually set up a solution for the optimization problem. Who set up the problem properly?

A. Surfy: Minimize 10-x/5 + √9+x²/3 when 0 ≤ x ≤ 10.
B. Sunny: Minimize 10-x/3 + √9+x²/5 when 0 ≤ x ≤ 10.
C. Boogie: Minimize x+5 + √9+x²/3 when 0 ≤ x ≤ 10.
D. Windy: Minimize 5 ∙ (10 – x) +3 ∙ √9+x² when 0 ≤ x ≤ 10.
E. Wavy : Minimize 5∙x +3. √9+x² when 0 ≤ x ≤ 10.

Answer 1

The correctly set up optimization problem to minimize the time it takes for a surfer to reach an island from a surfing school combines the time walking and time paddling. Only Surfy's setup, option A, reflects the correct formulation of the travel time function. It takes into account walking at 5 mph and paddling at 3 mph correctly.

Step-by-step explanation:

The question is asking for the setup of an optimization problem where a surfer wishes to minimize the time taken to reach a resort on an island from the nearest surfing school. The surfer has the option to walk along the shore and then paddle to the island. The question is to determine how far down the shoreline from point P the surfer should begin to surf in order to minimize the time taken.

To set up the correct optimization problem, we must consider the distance walked and the distance paddled separately since they occur at different speeds. Walking occurs at a speed of 5 miles per hour and paddling occurs at a speed of 3 miles per hour. The total distance along the shoreline from the surfing school to point P is 10 miles.

Let's define x as the distance walked along the shore from point P. The distance left to walk will then be (10 - x) miles. This distance is walked at 5 miles per hour. After walking, the surfer has to paddle a distance represented by the hypotenuse of a right triangle where one leg is the remaining distance on the shore, (10 - x), and the other leg is the distance across the water to the island, a constant 3 miles. The hypotenuse can be calculated using the Pythagorean theorem as √(9 + x²). The surfer paddles this distance at a speed of 3 miles per hour.

The correct setup to minimize the total travel time will combine the time taken walking and paddling, which results in this function:

Minimize (10 - x)/5 + √(9 + x²)/3 subject to the constraint that 0 ≤ x ≤ 10.

This function represents the total time as the sum of the time walking (distance over speed) and the time paddling (distance over speed).

From the given options, the one who set up the problem correctly is Surfy, represented by option A: Minimize 10-x/5 + √9+x²/3 when 0 ≤ x ≤ 10.