Math Problem Statement

You are a lifeguard and spot a drowning child 40 meters along the shore and 50 meters from the shore to the child. You run along the shore for a while and then jump into the water, swimming from there directly to the child. You can run at a rate of 3 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should you run before jumping into the water in order get to the child in the least amount of time?

Hint: Label the distance from the origin where you jump in the water x. Can you come up with a formula in terms of x for the distance you swim in the water? You are minimizing time so you need to come up with a formula for the total time to go along the shore plus the time swimming)

Round your answer to three decimal places

Solution

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Math Problem Analysis

Mathematical Concepts

Optimization
Distance Formula
Derivatives

Formulas

Distance formula
Time formula

Theorems

Derivative rules

Suitable Grade Level

Grades 11-12