Draw a circular pool with a lifeguard on one side and a drowner on the opposite side. The lifeguard swims with velocity v and runs around the rest of the pool with velocity w = 10v... Draw a circular pool with a lifeguard on one side and a drowner on the opposite side. The lifeguard swims with velocity v and runs around the rest of the pool with velocity w = 10v. If the swim direction is at angle θ with the direct line, choose θ to minimize and maximize the arrival time. Read more

Steps to Solve

1. Set Up the Problem

Let the radius of the circular pool be ( r ). The lifeguard is at point A (on the edge of the pool) and the drowner is at point B (directly opposite A). The angle ( \theta ) is the angle that the lifeguard swims at to reach the line connecting A and B.

2. Determine Distances

Using geometry, when the lifeguard swims, the distance to the drowner can be calculated using trigonometrical functions. The distance swum in the water is given as: $$ d_s = r \cdot \cos(\theta) $$

The remaining distance around the pool when running is: $$ d_r = r \cdot (1 - \cos(\theta)) $$

3. Calculate Times

The time taken to swim and run can be expressed as:

• Time to swim: $$ t_s = \frac{d_s}{v} = \frac{r \cdot \cos(\theta)}{v} $$

• Time to run: $$ t_r = \frac{d_r}{w} = \frac{r \cdot (1 - \cos(\theta))}{10v} $$

4. Total Time Function

The total time ( T ) to reach the drowner is: $$ T = t_s + t_r = \frac{r \cdot \cos(\theta)}{v} + \frac{r \cdot (1 - \cos(\theta))}{10v} $$

5. Simplify the Total Time

Factoring out constants: $$ T = \frac{r}{v} \left( \cos(\theta) + \frac{(1 - \cos(\theta))}{10} \right) $$

6. Finding Minimum and Maximum

To minimize or maximize ( T ), differentiate with respect to ( \theta ): $$ \frac{dT}{d\theta} = 0 $$ Solving this gives the critical points for ( T ).

7. Test Critical Points

Identify the critical points and check their values at the boundary (where ( \theta = 0 ) and ( \theta = 90^\circ )) to determine the minimum and maximum times.