Problem 49 - Precalculus: Functions and Graphs

Question

Cargo winch A large winch of diameter 3 feet is used to hoist cargo, as shown in the figure. (a) Find the distance the cargo is lifted if the winch rotates through an angle of radian measure \(7 \pi / 4\) (b) Find the angle (in radians) through which the winch must rotate in order to lift the cargo \(d\) feet.

Step-by-Step Solution

Answer

(a) Cargo is lifted \(\frac{21\pi}{8}\) feet. (b) Rotate through \(\frac{2d}{3}\) radians to lift \(d\) feet.

Step 1: Find Circumference of the Winch

The circumference of a circle is given by the formula \( C = \pi \times d \) where \( d \) is the diameter of the circle. For the winch with a diameter of 3 feet, the circumference \( C = \pi \times 3 = 3\pi \) feet.

Step 2: Calculate Distance Lifted Using Arc Length Formula

The arc length \( L \) of a circle corresponding to a central angle \( \theta \) is given by the formula \( L = r \theta \), where \( r \) is the radius. Alternatively, since \( r = \frac{d}{2} \), \( L \) can also be expressed as \( \frac{d\theta}{2} \). Thus, the distance lifted when the winch rotates through an angle \( \frac{7\pi}{4} \) radians is: \[ L = 3\pi \times \frac{7\pi}{4} \div 2 = \frac{3\times7\pi}{8} = \frac{21\pi}{8} \] feet.

Step 3: Solve for Required Angle Rotation for Given Distance

Let's rearrange the formula \( L = r \theta \) to solve for \( \theta \): \( \theta = \frac{2L}{d} \). To lift the cargo \( d \) feet, substitute \( L = d \) and \( r = 1.5 \) feet (since the radius is half of the diameter), to get: \[ \theta = \frac{d}{1.5} = \frac{2d}{3} \] radians.

Key Concepts

Angle of Rotation

Understanding the angle of rotation is crucial when dealing with circular movements like the rotation of a winch. When a circle or a part of it rotates, it sweeps out an angle. In the context of the winch problem, the angle of rotation tells us how much the winch turns, which directly affects the distance the cargo is lifted.

The angle is often measured in radians, which provides a direct relationship to the arc length on the circle's circumference. For example, a full rotation around a circle equates to an angle of \(2\pi\) radians. Therefore, knowing the angle in radians allows us to precisely calculate the path along the circle — or arc length — traversed by the winch's rope during lifting.

Circle Circumference

The circumference of a circle is like the perimeter in a polygon — it's the distance around the circle. Calculating the circumference is foundational when you're interested in how far a point travels around a circle.

The formula to determine a circle's circumference is simple:

• \( C = \pi \times d \)

where \(d\) is the diameter. In our winch scenario, the diameter is 3 feet, so the circumference is \(3\pi\) feet. This measure is critical because it allows you to relate the angle through which a circle turns to a corresponding linear distance. Whenever an angle is given, you can determine how far along the circumference an object moves, which was the basis for calculating how much the winch lifts the cargo.

Radian Measure

Radian measure is essential for connecting angles to arc lengths on circles. Unlike degrees, radians give a natural way of expressing angles, especially in mathematics involving periodic functions like trigonometry.

One radian is the angle formed when the arc length is equal to the circle's radius. To visualize this, imagine wrapping the radius along the circumference of the circle.

• The entire circle is \(2\pi\) radians since the circumference \(C = 2\pi r\).
• Radians provide an intrinsic link to the circumference and are used to calculate arc lengths with the formula \( L = r \theta \).

Understanding radians lets you quickly assess how far an object travels along a circle's edge, as seen in the winch problem where \(\frac{7\pi}{4}\) radians determined the lift distance.

Diameter of a Circle

The diameter of a circle is the longest straight line that can be drawn through the center, connecting two points on the circle's boundary. While simple conceptually, it plays a vital role in circular geometry.

The diameter is twice the length of the radius, represented as \(d = 2r\). It's especially useful because many formulas rely on knowing the diameter or radius, such as:

• Circumference \(C = \pi d\)
• Circle area \(A = \pi r^2\)

In the context of our winch exercise, knowing the diameter allowed us to determine the winch's circumference, which was then used alongside the given angle to find the arc length — the distance the cargo would be lifted.