Problem 81 - Precalculus Mathematics for Calculus

Question

A winch of radius \(2 \mathrm{ft}\) is used to lift heavy loads. If the winch makes 8 revolutions every \(15 \mathrm{s}\), find the speed at which the load is rising. (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The load is rising at approximately 6.7 ft/s.

Step 1: Convert revolutions to distance traveled

First, we determine the distance the load travels per revolution. Since the load is being lifted by the winch, it follows the circumference of the winch per revolution. The circumference of a circle (or in this case, the winch) is given by the formula \( C = 2\pi r \), where \( r \) is the radius. With a radius of \( 2 \) ft, the circumference is: \[ C = 2 \pi \times 2 = 4\pi \text{ ft} .\] Therefore, each revolution lifts the load \( 4\pi \) feet.

Step 2: Calculate distance for multiple revolutions

Next, we find out how much distance the load is lifted in 8 revolutions. We multiply the circumferential distance per revolution by the number of revolutions: \[ \text{Distance} = 8 \times 4\pi = 32\pi \text{ ft}. \] Thus, in 8 revolutions, the load is lifted \( 32\pi \) feet.

Step 3: Determine the time duration for the revolutions

According to the problem, these 8 revolutions occur over 15 seconds. This information will be used to calculate the speed of the load in the next step.

Step 4: Calculate the speed of the rising load

Speed is defined as the distance covered divided by the time taken. Using the distance from Step 2 and the time from Step 3, the speed at which the load is rising can be calculated as: \[ \text{Speed} = \frac{32\pi \text{ ft}}{15 \text{ s}}. \] Simplifying gives us: \[ \text{Speed} \approx 6.7 \text{ ft/s}. \]

Key Concepts

Trigonometry and Circular Motion

In precalculus, trigonometry and circular motion are two intertwined topics, especially when dealing with rotational systems like winches. Trigonometry helps us understand the geometric properties of circles, which can be applied to the motion of objects around a circular path. In the context of this problem, the winch follows a circular motion pattern, which can be analyzed using concepts of trigonometrical calculations and geometry.

When a wheel or a winch makes a full rotation, it covers a distance equal to its circumference. The winch mentioned in the exercise has a radius of 2 feet, thus applying the formula for circumference, we calculate:

• Circumference: \( C = 2\pi r = 4\pi \text{ ft} \)
• The load travels \( 4\pi \text{ ft} \) for every complete revolution.

These calculations highlight the relation between radius and circumference, a core aspect of trigonometry and circular motion. The number of revolutions provides a clear method to understand how far the load has been lifted by the winch using trigonometric concepts.

Understanding Circular Motion

Circular motion not only involves rotational distances but also the time in which these rotations occur. This aspect lands us in the realm of physics, where circular motion requires understanding speed and angular velocity. When looking at how quickly the load rises on a winch, we see it as a combination of angular velocity and linear speed.

Angular velocity tells us how fast something rotates around a circle, while linear speed describes how fast it's moving along that circular path. In this exercise, the winch makes 8 revolutions in 15 seconds. Translating these revolutions into a linear path allows us to compute the speed of the load. The winch lifts the load by:

• Distance in 8 revolutions: \( 32\pi \text{ ft} \)
• Time: 15 seconds

This relationship is crucial for understanding the principles of circular motion and its correlation with linear speeds in problems like the one at hand.

Speed and Calculus Concepts

Calculus plays a significant role in understanding motion, and it's used to determine variables like speed and acceleration. Even though this problem primarily uses basic arithmetic for the calculations, the underlying concepts connect deeply with calculus.

Here, the load's speed is found by dividing the total distance by time, forming the basis of calculus concepts like derivatives where the rate of change is measured. Speed can be thought of as how quickly the position of the load is changing with time. Calculus also introduces us to more complex ideas, such as how speed might change if acceleration is involved.

• Load lifted distance: \( 32\pi \text{ ft} \)
• Time duration: 15 seconds
• Calculated speed: \( \text{Speed} = \frac{32\pi \text{ ft}}{15 \text{ s}} \approx 6.7 \text{ ft/s} \)

Understanding this simple calculation sets the foundation for deeper explorations into calculus where concepts like limits, rates of change, and differential equations become pivotal.