Problem 327 - Calculus Volume 1

Question

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

Step 1: Understand the Geometry of the Problem

The pool is circular with a diameter of \(40\, \text{m}\). This means the radius \(r\) is \(20\, \text{m}\). The lifeguard swims diagonally across a region of the pool and runs along the circular edge to reach point \(C\). The distance for the diagonal swim is related to the "cord" of the selection arc defined by the swim angle \(\theta\).

Step 2: Determine Swimming Distance and Speed

The swimmer crosses the pool directly opposite by the angle \(\theta\), breaking up the circular path. The swimming distance \(d_1\) can be visualized as a straight line defined as: \[ d_1 = 2r \sin \left(\frac{\theta}{2}\right) = 40\sin\left(\frac{\theta}{2}\right) \] where \(r\) is the radius.

Step 3: Determine Running Distance and Speed

The lifeguard completes the rest of the distance by running along the circular edge of the pool. The running distance \(d_2\) is the arc segment \(\theta\), calculated as: \[ d_2 = r\theta = 20\theta \] Note that the total angle in radians of a full circle is \(2\pi\), he covers \(\theta\) swimming and \(2\pi - \theta\) running.

Step 4: Calculate Total Time as a Function of \(\theta\)

The total time \(T(\theta)\) to cover both distances can be represented by combining both the swimming and running durations:\[ T(\theta) = \frac{d_1}{v} + \frac{d_2}{w} \]Substitute for the distances:\[ T(\theta) = \frac{40\sin\left(\frac{\theta}{2}\right)}{v} + \frac{20(2\pi - \theta)}{3v} \]

Step 5: Simplify the Function T(\theta)

Combine similar terms and simplify the equation,\[ T(\theta) = \frac{40\sin\left(\frac{\theta}{2}\right)}{v} + \frac{40\pi}{3v} - \frac{20\theta}{3v} \].The function \(T(\theta)\) now succinctly defines the total time taken in terms of \(\theta\).

Key Concepts

Circular Motion

Circular motion occurs when an object moves along a circular path. In the case of the lifeguard in the circular pool, he experiences circular motion whenever he runs along the edge of the pool. This motion is not linear, but rather follows the circular boundary of the pool.

Key elements to remember about circular motion include:

• The path is defined by the radius of the circle, which is half of the diameter. Here, the diameter is 40 m, so the radius is 20 m.
• As the lifeguard runs around the pool, he covers a certain arc length. This arc length can be calculated by multiplying the radius with the angle in radians that the arc subtends at the center of the circle.
• In full circular motion, the total angle around the circle is equal to 2π radians.

The lifeguard needs to combine both swimming and running to minimize time, therefore evaluating how circular motion plays a role in his running part is significant to optimizing his path.

Angle of Swimming

The angle of swimming, represented by \(\theta\), is crucial to determining the quickest path the lifeguard can take. This angle defines the path the lifeguard swims across the pool and impacts how much of the distance is covered by swimming versus running.

Here’s how the angle of swimming affects the scenario:

• The swimming distance \(d_1\) is calculated using the formula \(d_1 = 40\sin\left(\frac{\theta}{2}\right)\). This shows that the swim path is dependent on the sine of half the angle \(\theta\).
• Choosing the right angle \(\theta\) will balance your swimming and running times. Less swimming time means longer running, and vice versa.
• The total angle in degrees or radians not only tells you how much he should swim directly but also how much he will need to circumvent by running.

Selecting an optimal angle \(\theta\) is key to reducing the total time the lifeguard takes to reach the drowning person, therefore understanding and adjusting this angle with regard to speed is a critical aspect of the problem.

Speed and Velocity in Physics

Speed and velocity are fundamental concepts in physics, both relevant in this problem. They define how fast and in what direction the lifeguard moves, either when swimming or running.

Important points concerning speed and velocity:

• Speed is the magnitude of velocity and is a scalar quantity. It shows how fast the lifeguard moves, but not the direction.
• Velocity is a vector quantity that shows both speed and the direction of movement.
• The lifeguard swims at speed \(v\) and runs at speed \(w = 3v\). This implies the running speed is three times faster than swimming speed due to the nature of motion outside the water being less resistant than in water.

Understanding the difference between speed and velocity helps in visualizing the problem. The ability to run faster than swim influences the strategy to optimize the path taken. The lifeguard needs to balance the quicker running speed with longer running distances to minimize overall time, thus both speed and velocity are key to optimizing the solution.