Question

gerardo is a lifeguard and spots a drowning child 40 meters along the shore and 50 meters from the shore to the child. gerardo runs along the shore for a while and then jumps into the water and swims from there directly to the child. gerardo can run at a rate of 4 meters per second and swim at a rate of 1 meter per second. how far along the shore should gerardo run before jumping into the water in order to save the child? round your answer to three decimal places.

Explanation:

Formulate the total time function

$$ T(x) = \frac{x}{4} + \frac{\sqrt{(40 - x)^2 + 50^2}}{1} $$

Find the derivative and set to zero

$$ T'(x) = \frac{1}{4} + \frac{1}{2\sqrt{(40 - x)^2 + 2500}} \cdot 2(40 - x) \cdot (-1) = 0 $$

$$ \frac{1}{4} = \frac{40 - x}{\sqrt{(40 - x)^2 + 2500}} $$

Solve for the optimal running distance

$$ (40 - x)^2 + 2500 = 16(40 - x)^2 $$

$$ 15(40 - x)^2 = 2500 $$

$$ 40 - x = \frac{50}{\sqrt{15}} \approx 12.910 $$

$$ x = 40 - \frac{50}{\sqrt{15}} \approx 27.090 $$

Answer:

Gerardo is a lifeguard and spots a drowning child 40 meters along the shore and 50 meters from the shore to the child. Gerardo runs along the shore for a while and then jumps into the water and swims from there directly to the child. Gerardo can run at a rate of 4 meters per second and swim at a rate of 1 meter per second. How far along the shore should Gerardo run before jumping into the water in order to save the child? Round your answer to three decimal places.
<blank>27.090</blank>