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.
Understand the Problem
The question is asking us to analyze a scenario involving a circular pool, where a lifeguard and a drowner are positioned at opposite sides. We need to determine the angle θ that the lifeguard should swim at to minimize and maximize the time it takes to reach the drowner, given different velocities for swimming and running.
Answer
Minimum time at \( \theta = 0 \); maximum time at \( \theta = 90^\circ \).
Answer for screen readers
To minimize the arrival time, the lifeguard should swim directly towards the drowner (( \theta = 0 )). To maximize the arrival time, the lifeguard should swim perpendicularly to the line towards the drowner (( \theta = 90^\circ )).
Steps to Solve
- 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.
- 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: $$ d_r = r \cdot (1 - \cos(\theta)) $$
- Calculate Times
The time taken to swim and run can be expressed as:
-
Time to swim: $$ t_s = \frac{d_s}{v} = \frac{r \cdot \cos(\theta)}{v} $$
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Time to run: $$ t_r = \frac{d_r}{w} = \frac{r \cdot (1 - \cos(\theta))}{10v} $$
- Total Time Function
The total time ( T ) to reach the drowner is: $$ T = t_s + t_r = \frac{r \cdot \cos(\theta)}{v} + \frac{r \cdot (1 - \cos(\theta))}{10v} $$
- Simplify the Total Time
Factoring out constants: $$ T = \frac{r}{v} \left( \cos(\theta) + \frac{(1 - \cos(\theta))}{10} \right) $$
- 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 ).
- 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.
To minimize the arrival time, the lifeguard should swim directly towards the drowner (( \theta = 0 )). To maximize the arrival time, the lifeguard should swim perpendicularly to the line towards the drowner (( \theta = 90^\circ )).
More Information
- In this scenario, swimming directly towards the drowner minimizes the distance traveled in the water, while swimming perpendicularly maximizes the time taken due to the increased distance that must be swum.
Tips
- Confusing the distances swum and run as being equal.
- Forgetting to account for the velocities when calculating times.
- Not considering the trigonometric relationships correctly for angles.