Problem 49 - Algebra and Trigonometry with Analytic Geometry

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) Distance lifted is \(\frac{21\pi}{8}\) feet; (b) Angle is \(\frac{2d}{3}\) radians.

Step 1: Understanding the Circumference of the Winch

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

Step 2: Calculate Distance Lifted (Part a)

The distance the cargo is lifted is equivalent to the arc length, which can be calculated using the formula \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. Since the diameter is 3 feet, the radius \( r = \frac{3}{2} \). The given angle \( \theta = \frac{7\pi}{4} \). Thus, the distance lifted \( s = \frac{3}{2} \times \frac{7\pi}{4} = \frac{21\pi}{8} \) feet.

Step 3: Calculate the Angle for Lifting a Specific Distance (Part b)

To find the angle \( \theta \) in radians needed to lift the cargo \( d \) feet, use the arc length formula rearranged: \( \theta = \frac{s}{r} \). With the radius \( r = \frac{3}{2} \) and the desired distance \( s = d \), the angle is \( \theta = \frac{d}{\frac{3}{2}} = \frac{2d}{3} \).

Key Concepts

Arc Length

Let's talk about arc length, a fascinating topic in circular motion. When an object, like a winch, moves around a circle, the distance it travels along the edge of the circle is known as the arc length. This can be thought of as "how far" the object has moved along the circle's path. To calculate the arc length, we use the formula:

• \( s = r\theta \)

where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the angle in radians through which the circle has rotated.Arc length is a simple yet powerful concept. It helps us understand how far something has moved in circular motion based on how much it has rotated. This is especially useful in systems involving wheels, pulleys, or winches.

Radian Measure

Radian measure is a fundamental way of expressing angles. Unlike degrees, which divide a circle into 360 parts, radians use the circle itself for measurement. One full rotation around a circle is equal to \( 2\pi \) radians.Why use radians? Here are a few reasons:

• They simplify many mathematical formulas, such as those involving arc length and sector area.
• They naturally relate a circle's geometry to its algebraic properties.

An angle in radians is defined as the ratio of the arc length to the radius of the circle:

• \( \theta = \frac{s}{r} \).

This makes radians particularly useful in physics and engineering, where circular motion is common.

Circumference

The circumference of a circle is another vital concept. It represents the total distance around a circle. You can think of it as the circle's perimeter. To calculate the circumference, you use the formula:

• \( C = \pi d \)

where \( d \) is the diameter of the circle.If you know the radius, you can also express circumference as:

• \( C = 2\pi r \)

In the context of the winch, knowing the circumference allows you to relate the rotation of the winch directly to the amount of cargo lifted. The 3 feet diameter winch has a circumference of \( 3\pi \) feet, meaning this is the distance the winch covers in one complete turn.

Geometry of Circles

The geometry of circles is a rich field, encompassing various properties and theorems. A circle is defined by all points equidistant from a central point. Key elements include:

• Radius: The distance from the center to any point on the circle. It is half the diameter.
• Diameter: A line that passes through the center and spans the width of the circle. It is twice the radius.
• Arc: A section of the circle's circumference, defined by two endpoints.
• Sector: The area enclosed by an arc and two radii.

Understanding these elements can help you solve various problems involving circles, such as finding areas, lengths, and angles. This foundational knowledge supports more advanced studies in mathematics and physics, particularly those involving motion and force.