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 needs to reach a drowner. The lifeguard has two velocities: one for swimming and one for running. We need to determine the angle at which the lifeguard should swim to either minimize or maximize the time it takes to arrive at the drowner.
Answer
The angle $\theta$ can be solved by optimizing the total time function derived from swimming and running distances, noting that a straight swim minimizes time.
Answer for screen readers
The angles $\theta$ necessary to minimize or maximize the arrival time can be found by solving the equation derived from: $$ \frac{dT}{d\theta} = 0 $$
After performing this analysis, the specific angles will depend on the values in the context of the lifeguard's speed relative to the distance.
Steps to Solve
- Identify the scenario parameters
Let the radius of the circular pool be $R$. The lifeguard starts at point A (one side of the pool) and needs to reach point B (the opposite side where the drowner is located). The angle $\theta$ will be formed between the line connecting points A and B and the path the lifeguard swims.
- Determine distances for swimming and running
The distance the lifeguard swims (along the chord) can be calculated using trigonometric relations. The swim distance $D_s$ is given by: $$ D_s = 2R \sin\left(\frac{\theta}{2}\right) $$
The remaining distance the lifeguard runs around the edge of the pool (from point C back to B) can be found using: $$ D_r = R \left(\pi - \theta\right) $$
- Calculate times for swimming and running
Next, compute the time taken for swimming $T_s$ and running $T_r$:
The time to swim is given by: $$ T_s = \frac{D_s}{v} = \frac{2R \sin\left(\frac{\theta}{2}\right)}{v} $$
The time to run is: $$ T_r = \frac{D_r}{w} = \frac{R \left(\pi - \theta\right)}{10v} $$
- Combine times to find total time
The total time $T$ taken to reach the drowner is the sum of the swimming and running times: $$ T = T_s + T_r = \frac{2R \sin\left(\frac{\theta}{2}\right)}{v} + \frac{R \left(\pi - \theta\right)}{10v} $$
- Minimize and maximize the total time
To find the angle that minimizes or maximizes the time, take the derivative of $T$ with respect to $\theta$ and set it to zero: $$ \frac{dT}{d\theta} = 0 $$
Once you find the critical points, use the second derivative test or evaluate their behavior to determine if they minimize or maximize the time.
The angles $\theta$ necessary to minimize or maximize the arrival time can be found by solving the equation derived from: $$ \frac{dT}{d\theta} = 0 $$
After performing this analysis, the specific angles will depend on the values in the context of the lifeguard's speed relative to the distance.
More Information
In this scenario, the angle $\theta$ that minimizes the arrival time typically results in the lifeguard swimming directly towards the drowner, reducing unnecessary distance traveled. Conversely, maximizing the time can occur when the lifeguard swims at an angle that extends the travel distance more than necessary.
Tips
- A common mistake is neglecting to consider both swimming and running times as a total time expression.
- Another mistake is miscalculating the lengths of the swimming and running distances based on the angle $\theta$.
- Failing to correctly apply differentiation and test for minima and maxima.
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