Problem 173 - Calculus Volume 1

Question

The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of \(120^{\circ},\) how many inches does it move? Approximate to the nearest whole inch.

Step-by-Step Solution

Answer

The wheel moves approximately 42 inches.

Step 1: Understand the Concept

When a wheel rotates, the distance it covers on the ground is equal to the arc length of the circle that's traversed by the point on the circumference. The relationship between the angle of rotation and the arc length is given by the formula for arc length: \[ L = \frac{\theta}{360^{\circ}} \times 2 \pi r \], where \( L \) is the arc length, \( \theta \) is the angle of rotation in degrees, and \( r \) is the radius of the circle.

Step 2: Determine the Radius

The diameter of the wheel is given as 40 inches. The radius \( r \) is half of the diameter, so: \[ r = \frac{40}{2} = 20 \text{ in} \].

Step 3: Convert the Angle to Fraction

The wheel rotates through an angle of \(120^{\circ}\). Convert this angle to a fraction of a full rotation (360 degrees): \[ \frac{120}{360} = \frac{1}{3} \].

Step 4: Calculate the Arc Length

Using the formula for arc length: \[ L = \frac{1}{3} \times 2 \pi \times 20 \]. First, calculate \(2 \pi \times 20\): \(40 \pi\). Then multiply by \(\frac{1}{3}\): \[ L = \frac{1}{3} \times 40 \pi \approx \frac{125.66}{3} \approx 41.89 \text{ inches} \].

Step 5: Round to Nearest Whole Inch

The calculated arc length \(41.89\) inches should be rounded to the nearest whole number. Thus, the wheel moves approximately 42 inches.

Key Concepts

Circle Circumference

The circumference of a circle is the total distance around the circle. It is an important value used to understand different aspects of circular paths, like what a wheel travels on the ground.
The formula to calculate the circumference is given by:

• Circumference of a circle, \[ C = 2 \pi r \], where \( r \) is the radius.

The circumference helps us to understand the full path a wheel would cover if it rotated entirely around once. In this case, the wheel has a radius of 20 inches, so the full circumference would be \( 40 \pi \) inches or approximately 125.66 inches. Knowing this, every partial rotation (like 120 degrees in our example) will cover a fraction of this total circumference.

Angle of Rotation

An angle of rotation indicates how far a wheel or any circular object has turned from its original position. We measure this angle in degrees.
A full rotation of a circle is 360 degrees. In exercises like the one presented, it's crucial to calculate what fraction of the full circle the wheel has completed when given an angle, such as 120 degrees.

• To convert the given angle into the fraction of a circle: \[ \text{Fraction of rotation} = \frac{\text{angle}}{360^{\circ}} \]
• For 120 degrees, the fraction is: \( \frac{120}{360} = \frac{1}{3} \)

Understanding this fraction aids in calculating how much of the circle’s path (the circumference) the wheel has traversed. In our case, since the wheel rotates through \( \frac{1}{3} \) of the circle, it covers \( \frac{1}{3} \) of the total circumference.

Diameter and Radius

The diameter and radius are key measurements in understanding circles. The diameter is the straight line passing through the center of the circle from one edge to the other, while the radius is half of that diameter.

• Formula connecting radius and diameter: \[ Diameter = 2 \times Radius \]
• In this exercise, the diameter is 40 inches. Thus, the radius is: \[ r = \frac{40}{2} = 20 \text{ inches} \]

These measurements are essential because they directly relate to other circle properties like the circumference and arc length. Knowing either the diameter or radius allows for further calculations necessary for solving problems involving circular motion.

Geometry Concepts

The problem of finding the distance a wheel moves when it rotates involves various geometry concepts. Understanding the relationships among angles, curves, and radii is foundational.

• An essential geometry concept is arc length, which is part of the circumference that the angle of rotation spans on the circle. The arc length formula is given by: \[ L = \frac{\theta}{360^{\circ}} \times 2 \pi r \].
• In this formula, \( \theta \) is the rotational angle, \( r \) is the radius, and \( L \) is arc length.
• The formula uses the fraction of the circle that the rotation corresponds to, multiplying by the full circumference to find the arc length.

Understanding these geometrical principles supports solving practical problems involving circular movements, like how far a wheel rolls.