Problem 43 - Algebra 2 and Trigonometry

Question

A wheel whose radius measures 16 inches is rotated. If a point on the circumference of the wheel moves through an arc of 12 feet, what is the measure, in radians, of the angle through which a spoke of the wheel travels?

Step-by-Step Solution

Answer

The angle is 9 radians.

Step 1: Convert Units

First, we need to ensure that both the radius and arc length are in the same units. Given that the radius is 16 inches and the arc is measured in feet, let's convert the arc to inches. Since 1 foot equals 12 inches, 12 feet is equivalent to \(12 \times 12 = 144\) inches.

Step 2: Use the Arc Length Formula

The formula for the arc length \(s\) of a circle is given by \(s = r \times \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. In this case, \(s = 144\) inches and \(r = 16\) inches. We need to solve for \(\theta\).

Step 3: Solve for the Angle

Rearrange the formula to solve for \(\theta\): \(\theta = \frac{s}{r}\). Substitute the known values: \(\theta = \frac{144}{16}\).

Step 4: Calculate the Angle

Calculate \(\theta\) by dividing 144 by 16: \(\theta = 9\).

Step 5: Conclusion

The measure of the angle in radians through which a spoke of the wheel travels is 9 radians.

Key Concepts

Radians

Radians are a unit of measurement used to describe angles. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. One complete revolution around a circle is equivalent to 2π radians. This is because the circumference of the circle is the distance covered when the radius travels once around the circle. Here's a neat way to think about it:

• 1 radian is the angle created when the arc length equals the radius of the circle.
• Since the circumference of a circle is given by the formula \(2\pi r\), a full circle corresponds to \(2\pi\) radians.
• It's a natural way to measure angles, especially in mathematical contexts, because it directly connects the angle to the length of the arc subtended.

Using radians simplifies many mathematical equations and calculations, especially those involving trigonometric functions.

Circle

A circle is an essential shape in geometry, defined as the set of all points in a plane that are a fixed distance from a given point, known as the center. This fixed distance is called the radius. Key characteristics include:

• The diameter, which is twice the length of the radius, stretching across the circle from one point to another.
• The circumference, which is the total distance around the circle. Calculated by the formula \(2\pi r\).
• An arc, which is a portion of the circumference, and can be thought of like a "curved edge" of the circle.

In problems involving motion around circles, such as wheels or gears, understanding the relationships between these parts is crucial. The concepts of radius, diameter, and circumference all play a significant role in calculating distances and rotations.

Angle Measurement

Angle measurement is a way to quantify the rotation needed to bring one ray or line to another. In the context of circles, it's all about understanding how much of a turn an object has made.In our exercise, \(\theta\) represents the measure of the angle in radians. Calculating angle measurements often involves:

• Understanding the arc length \(s\).
• Knowing the radius \(r\) of the circle.
• Utilizing the arc length formula \(s = r \times \theta\).

This formula is fundamental because it links the distance around the arc to the radius and the angle, allowing us to solve for any one of these if the other two are known. Measuring angles in radians is often preferred in many fields, including engineering and physics, due to its direct relationship with these linear parameters.

Unit Conversion

Unit conversion is crucial when working with different units of measurement. In the given exercise, both the radius and arc length must be in the same units for calculations to work smoothly. To convert, pay attention to:

• What units you start with (e.g., inches, feet).
• The units needed for consistency.
• The conversion factors (e.g., 1 foot = 12 inches).

In this problem, the arc length was initially given in feet, while the radius was in inches. Converting the arc length into inches was necessary to use the arc length formula effectively without errors. Consistent units ensure clarity and accuracy in calculations, preventing mix-ups and erroneous results in any math problem.