{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>


a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.

Verified step by step guidance

1. First, identify the variables involved in the problem. Let x be the distance from point Q along the shore where you stop running and start swimming.
2. Next, express the running distance in terms of x. Since you start at point P, which is 50 meters from Q, the running distance is 50 - x meters.
3. Now, express the swimming distance. The swimmer is 50 meters from point Q, so the swimming distance forms a right triangle with the shore. Use the Pythagorean theorem to find the swimming distance: \( \sqrt{x^2 + 50^2} \).
4. Determine the time taken for each segment of the journey. The running time is \( \frac{50 - x}{4} \) seconds, and the swimming time is \( \frac{\sqrt{x^2 + 50^2}}{2} \) seconds.
5. Combine these expressions to form the total travel time function T(x): \( T(x) = \frac{50 - x}{4} + \frac{\sqrt{x^2 + 50^2}}{2} \). This function represents the total time taken to reach the swimmer as a function of x.

Key Concepts

Optimization

Optimization is a fundamental concept in calculus that involves finding the maximum or minimum values of a function. In this context, we need to minimize the total travel time to reach the swimmer. This often requires setting up a function that represents the total time based on the distance run and the distance swum, and then using techniques such as differentiation to find critical points.

Distance and Speed Relationships

Understanding the relationship between distance, speed, and time is crucial for solving this problem. The basic formula, time = distance/speed, allows us to express the time taken to run and swim in terms of the distances involved. By breaking down the journey into running and swimming segments, we can create a function that accurately reflects the total time based on the chosen point along the shore.

Functions and Graphs

In calculus, functions represent relationships between variables, and graphs visually depict these relationships. For this problem, we will define a function T(x) that represents the total travel time as a function of the distance x from point Q. Analyzing this function, including its domain and behavior, is essential for determining the optimal point to switch from running to swimming.