Question

Shannon is a lifeguard and spots a drowning child 30 meters along the shore and 40 meters from the shore to the child. Shannon runs along the shore for a while and then jumps into the water and swims from there directly to the child. Shannon can run at a rate of 3 meters per second and swim at a rate of 1.1 meters per second. How far along the shore should Shannon run before jumping into the water in order to save the child? Round your answer to three decimal places.

Answer 1

Shannon should run along the shore for a distance of 0 meters before jumping into the water to save the child.

Step-by-step explanation:

Shannon should run along the shore for a distance that allows her to reach the child in the least amount of time. To find this distance, we can calculate the time it takes for Shannon to run and then swim to the child.

Let x be the distance Shannon runs along the shore before jumping into the water. The time it takes her to run this distance is x/3 seconds. After jumping into the water, Shannon will then swim the remaining 30 meters to the child at a rate of 1.1 meters per second. The time it takes her to swim this distance is 30/1.1 = 27.27 seconds.

The total time it takes Shannon to reach the child is the sum of the time it takes her to run and swim: x/3 + 27.27 seconds.

To minimize the total time, we need to minimize the time it takes to run and swim. Therefore, we need to minimize the function f(x) = x/3 + 27.27.

To find the minimum value of f(x), we can take the derivative of f(x) with respect to x and set it equal to zero:

f'(x) = 1/3 = 0

Solving for x, we find x = 0. Multiplying this value by the rate at which Shannon runs (3 meters per second), we get the distance Shannon should run before jumping into the water: 0 meters.