Problem 327 - Calculus Volume 1

Question

For the following problems, consider a lifeguard at a circular pool with diameter 40 \(\mathrm{m}\) . He must reach someone who is drowning on the exact opposite side of the pool, at position \(C .\) The lifeguard swims with a speed \(v\) and runs around the pool at speed \(w=3 v\) Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, \(\theta .\)

Step-by-Step Solution

Answer

The function for the total time is \(T(\theta) = \frac{40}{v} \sin\left(\frac{\theta}{2}\right) + \frac{20(\pi - \theta)}{3v}\).

Step 1: Understanding the Problem

Imagine a circular pool with diameter 40 m. The lifeguard is initially at one point (call it A) and needs to reach exactly the opposite point (call it C). The task is to determine the time to reach C as a function of the swim angle \(\theta\). The lifeguard can swim directly or take any angle \(\theta\) and continue running around the perimeter.

Step 2: Determine Swimming Path

When the lifeguard swims at an angle \(\theta\), he covers a direct path across a chord. The length of this chord can be given by the formula involving the circle's radius: \( L_s = 2R \sin\left(\frac{\theta}{2}\right) \), where \(R = 20\, \text{m}\) is the pool's radius. Thus, the swimming distance \(L_s = 40 \sin\left(\frac{\theta}{2}\right)\).

Step 3: Determine Running Path

After reaching the chord endpoint, the lifeguard runs along the circular edge to reach point C. The running angle forms the complement angle to \(\theta\), which is \(\pi - \theta\). The running distance \(L_r\) along the circle's edge is given by the arc length formula: \( L_r = R(\pi - \theta) = 20(\pi - \theta)\).

Step 4: Total Time Calculation

Define \(T(\theta)\) as the total time. The time taken to swim is given by \(\frac{L_s}{v} = \frac{40 \sin\left(\frac{\theta}{2}\right)}{v}\), and the time to run is \(\frac{L_r}{3v} = \frac{20(\pi - \theta)}{3v}\). Therefore, the total time function \(T(\theta)\) is: \[ T(\theta) = \frac{40 \sin\left(\frac{\theta}{2}\right)}{v} + \frac{20(\pi - \theta)}{3v} \].

Step 5: Simplify the Expression

The expression for total time \(T(\theta)\) can be simplified to: \[ T(\theta) = \frac{40}{v} \sin\left(\frac{\theta}{2}\right) + \frac{20(\pi - \theta)}{3v} \]. This function describes the total time taken depending on the chosen swim angle \(\theta\).

Key Concepts

Swimming Angle

The swimming angle, denoted as \( \theta \), is a critical component in this optimization problem. It represents the angle at which the lifeguard, standing at one point on a circular pool, chooses to swim directly towards where someone is in distress. The idea is that \( \theta \) determines both the path and length of the swimming segment, which is a chord across the pool. By manipulating \( \theta \), the lifeguard can manage the transition from swimming in a straight line to running around the pool's edge.

Choosing \( \theta \) wisely allows the lifeguard to minimize the time taken to reach the drowning person. It's important to note that the swimming angle affects the swim path by forming a chord, described mathematically by \( L_s = 40 \sin\left(\frac{\theta}{2}\right) \).

• An increase in \( \theta \) shortens the swim, but increases the run.
• A smaller \( \theta \) minimizes chord distance, but increases the arc to run.

By finding the optimal angle \( \theta \), the lifeguard efficiently balances the time spent swimming versus running.

Running Distance

After the lifeguard completes his swim across the chord of the pool, the next phase involves running around the pool's circumference to reach the opposite side. The running path is dictated by the complementary angle, \( \pi - \theta \), due to the circular shape of the pool.

The lifeguard's decision to swim at an angle \( \theta \) directly impacts how much further he needs to run. The calculation of this running distance, which extends along the circle's perimeter, relies on the arc length formula:

• \( L_r = 20(\pi - \theta) \)

The running segment is essential because the lifeguard runs at three times his swimming speed. As the running distance decreases with a larger \( \theta \), the trade-off is an increased swimming distance, highlighting the importance of the optimal balance between swimming and running.

Arc Length Formula

The arc length formula is a fundamental concept crucial to solving the optimization problem involving swimming and running. This mathematical formula helps in determining the distance along the circular path of the pool.

In our context, it's used to calculate the running distance, which depends on the angle \( \pi - \theta \). The formula verifies the arc segment with:

• \( L_r = R(\pi - \theta) \)
• With the pool's radius \( R \) being 20 meters, it simplifies to \( 20(\pi - \theta) \).

Understanding this formula means knowing how it quantifies the section of the perimeter the lifeguard will cover by running. It's the essential mathematical bridge between the physical pool and the time-based strategy the lifeguard employs, encompassing both running speed modifications and distance measures.

Time Function

The time function, \( T(\theta) \), expresses the total time the lifeguard takes to reach the drowning person as a function of the swim angle \( \theta \). It combines variables from both swimming and running segments to provide a singular value dependent on \( \theta \).

The expression for \( T(\theta) \) is given by:

• \( T(\theta) = \frac{40}{v} \sin\left(\frac{\theta}{2}\right) + \frac{20(\pi - \theta)}{3v} \)

Here, the formula splits into two parts:

• The swimming time: \( \frac{40 \sin\left(\frac{\theta}{2}\right)}{v} \)
• The running time: \( \frac{20(\pi - \theta)}{3v} \)

Each component accounts for how the lifeguard's speeds and the distances influence the total time. By optimizing \( \theta \), the lifeguard can minimize \( T(\theta) \), ensuring the fastest rescue. This time function reveals the mathematical relationship between physical angle and practical timing crucial in emergency responses.