Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the radius \(r\) of a circle with an arc length \(s\) and a central angle \(\theta\). Arc Length \(s\) 8 inches Central Angle \(\theta\) \(330^{\circ}\)

Step-by-Step Solution

Answer

The radius of the circle is approximately \(1.38\) inches.

Step 1: Convert degrees to radians

The first step is to convert the central angle from degrees to radians. The conversion factor is \(\frac{\pi}{180}\) radians per degree, so this would give us \[\theta = 330^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{11\pi}{6}\] radians.

Step 2: Rearrange the formula to solve for radius

We know that \[s = r\theta\] and we want to find \(r\), so we rearrange the formula so that we solve for \(r\): \[r = \frac{s}{\theta}\].

Step 3: Substitute values and solve

Now that we have converted our angle into radians and rearranged our formula to solve for \(r\), we can plug in the values of \(s\) and \(\theta\): \[r = \frac{8 \text{ inches}}{\frac{11\pi}{6}} = \frac{48}{11\pi} \approx 1.38 \text{ inches}\].

Key Concepts

Converting Degrees to Radians

Understanding how to convert degrees to radians is crucial for working with circles and angles in higher mathematics. To convert an angle from degrees to radians, use the conversion factor \(\frac{\pi}{180}\) radians per degree. The formula for this conversion is:ewlineewline \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \].ewlineewlineFor example, if you have an angle of \(330^\circ\), the conversion to radians would be:ewlineewline \[ \theta = 330^\circ \times \frac{\pi}{180^\circ} = \frac{11\pi}{6} \].ewlineewlineThis step is imperative since most mathematical functions that involve angles, like the arc length formula, use radians instead of degrees.

Arc Length Formula

The arc length formula is a fundamental tool in geometry that links the central angle of a circle in radians to the length of the corresponding arc. The formula is given by:ewlineewline \[ s = r\theta \],ewlineewlinewhere \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This formula assumes that the angle is measured in radians, which is why converting from degrees to radians is a necessary first step. By using this relationship, you can solve for any one of the three variables given the other two, making it a versatile equation for various geometry problems.

Solving for Radius

Solving for the radius of a circle when given the arc length and the central angle involves rearranging the arc length formula. To find the radius, you isolate \( r \) from the equation \( s = r\theta \):ewlineewline \[ r = \frac{s}{\theta} \].ewlineewlineWith this rearrangement, once you have the arc length \( s \) and the central angle \( \theta \) in radians, you simply divide the arc length by the angle. For example, if the arc length is 8 inches and the central angle is \( \frac{11\pi}{6} \) radians:ewlineewline \[ r = \frac{8 \text{ inches}}{\frac{11\pi}{6}} = \frac{48}{11\pi} \approx 1.38 \text{ inches} \].ewlineewlineThis calculation will provide you with the radius, which is a key element in understanding the geometry of a circle.

Central Angle and Arc Length

The central angle of a circle and the arc length have a direct relationship, as shown in the arc length formula. The central angle is the angle formed by two radii that enclose an arc on the circle's circumference. When given in radians, this angle tells us the proportion of the circle that the arc represents. By knowing either the radius and the arc length or the radius and the central angle, you can calculate the third measurement.ewlineewlineThis relationship is key to understanding circular motion, sector areas, and even aspects of trigonometry. It is also used frequently in engineering and design calculations where precision is required. Remember, always convert the central angle to radians when using it in formulas to ensure accuracy in your calculations.