Question

50 40 30 20 10 -10 10 10 20 30 40 Angela is a lifeguard and spots a drowning child 40 meters along the shore and 50 meters from the shore to the child. Angela runs along the shore for a while and then jumps into the water and swims from there directly to the child. Angela can run at a rate of 4 meters per second and swim at a rate of 1.1 meters per second. How far along the shore should Angela run before jumping into the water in order to save the child? Round your answer to three decimal places.

Answer 1

This is a simple question of find the minima of a function. We have to find the time taken by Angela to reach the child as a function of the distance along the shore Angela runs, and then minimize that function.

Let Angela run x meters along the shore. The the distance to the child from that point can be found using Pythagoras' theorem as:

$\small \sqrt{(40-x)^2+50^2}=\sqrt{x^2-80x+4100}$

Now, Angela runs along the shore at 4 meters per second, and swims at 1.1 meters per second, so the total time taken by Angela to run x meters and swim meters is given by

Vr? - 80x +4100

$\small \text{Time}=y(x)={x\over 4}+{\sqrt{x^2-80x+4100}\over 1.1}$

Now, we simply have to find the minimum of this function, y(x). Using a graphing calculator to draw a curve of the function y(x) we have

250 200, 150 100 50 (25.699, 53.702) 200 -150 -100 -50 0 50 100 150 200 250 -50

Thus, from the graph itself we can see that the function y(x) has a minimum value of 53.702 at x = 25.699

Thus, Angela should run 25.699 meters along the shore before jumping into the water, and the minimum time taken to reach the child is 53.702 seconds