Jeremiah is a lifeguard and spots a drowning child 40 meters along the shore and 70 meters from the shore to the child. Jeremiah runs along the shore for a while and then jumps into the water and swims from there directly to the child. Jeremiah 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 Jeremiah run before jumping into the water in order to save the child?  Round your answer to three decimal places.

This is a classic optimization problem that can be solved using calculus, specifically by applying the concept of minimizing time.

Let’s break down the problem and define key variables:
- Jeremiah runs along the shore a distance \( x \) (in meters) before jumping into the water.
- The total distance along the shore from Jeremiah’s starting point to the point directly opposite the child is 40 meters.
- The distance from the shore to the child (in the water) is 70 meters.
- Jeremiah can run at a speed of 4 meters per second and swim at a speed of 1.1 meters per second.

### Objective:
Minimize the total time it takes for Jeremiah to reach the child.

#### Step 1: Time running along the shore
The distance Jeremiah runs is \( x \) meters, and since his running speed is 4 m/s, the time spent running is:
\[
T_{\text{run}} = \frac{x}{4}
\]

#### Step 2: Time swimming in the water
The distance Jeremiah swims can be found using the Pythagorean theorem. The horizontal distance he swims is \( 40 - x \) (the remaining distance along the shore), and the vertical distance is 70 meters. So, the distance he swims is:
\[
d_{\text{swim}} = \sqrt{(40 - x)^2 + 70^2}
\]
Since his swimming speed is 1.1 m/s, the time spent swimming is:
\[
T_{\text{swim}} = \frac{\sqrt{(40 - x)^2 + 70^2}}{1.1}
\]

#### Step 3: Total time function
The total time, \( T(x) \), is the sum of the time running and the time swimming:
\[
T(x) = \frac{x}{4} + \frac{\sqrt{(40 - x)^2 + 70^2}}{1.1}
\]

#### Step 4: Minimize the time function
To find the optimal \( x \), we need to minimize the total time. We do this by taking the derivative of \( T(x) \) with respect to \( x \), setting it equal to zero, and solving for \( x \).

Let's go ahead and calculate this derivative and solve it.

#### Step 5: Solve for the optimal \( x \)

I'll now calculate the derivative and solve for \( x \).The optimal distance along the shore that Jeremiah should run before jumping into the water is approximately \( 19.978 \) meters, rounded to three decimal places.

Would you like more details on how this was calculated, or do you have any questions? 

Here are 5 related questions to consider:
1. How would the answer change if Jeremiah could swim faster, say at 2 meters per second?
2. What happens if the child is farther from the shore, say 100 meters instead of 70?
3. How much longer would it take Jeremiah to reach the child if he ran all the way to the point directly opposite the child before swimming?
4. Can you generalize this approach for any distances along the shore and from shore to child?
5. How does the time Jeremiah spends swimming compare to the time spent running at the optimal distance?

**Tip**: Always check the domain of your solution in optimization problems to ensure the result makes sense in the given context.

Jeremiah is a lifeguard and spots a drowning child 40 meters along the shore and 70 meters from the shore to the child. Jeremiah runs along the shore for a while and then jumps into the water and swims from there directly to the child. Jeremiah 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 Jeremiah run before jumping into the water in order to save the child? Round your answer to three decimal places.

Solve an optimization problem involving a lifeguard running and swimming to save a drowning child. Learn how to minimize the time using calculus, with key concepts like the Pythagorean theorem and derivatives. Jeremiah runs along the shore and jumps into the water to minimize rescue time. Suitable for students in Grades 11-12.

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