Problem 103 - 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 \(s\) 82 miles Central Angle \(\theta\) \(135^{\circ}\)

Step-by-Step Solution

Answer

The radius of the circle is approximately 34.96 miles.

Step 1: Convert central angle to radians

First convert the central angle from degrees to radians using the conversion factor \( \frac{\pi}{180} \). Thus \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} = 135^{\circ} \times \frac{\pi}{180} = \frac{3\pi}{4} \).

Step 2: Substitute the values into the formula

Now you can substitute the given arc length and the converted angle into the formula. It will look like this: \( r = \frac{s}{\theta} = \frac{82}{\frac{3\pi}{4}} \).

Step 3: Solve the equation

Finally, solve the equation for \(r\). It turns into a product: \( r = 82 \times \frac{4}{3\pi} \). This calculation will give you the radius of the circle.

Key Concepts

Central Angle to Radians Conversion

When you're dealing with circle measurements, understanding how to convert degrees to radians is key. The central angle of a circle, often given in degrees, needs to be converted to radians to use the arc length formula. Why is that? Well, radian measure is actually the standard unit of angular measurement in mathematics. It's directly related to the circle's radius and provides a way to express angles in a way that's intrinsically related to the circle's geometry.

To convert a central angle from degrees to radians, you can use the conversion factor \( \frac{\pi}{180} \). This factor is derived from the fact that a full circle is 360 degrees, which is equivalent to \(2\pi\) radians. Therefore, one degree is \( \frac{1}{360} \times 2\pi = \frac{\pi}{180} \) radians. Simply multiply your angle in degrees by \( \frac{\pi}{180} \). For instance, if the central angle is \(135^\circ\), the conversion to radians would look like this: \(135^\circ \times \frac{\pi}{180} = \frac{3\pi}{4}\).

This step is fundamental before moving on to find the arc length or radius of your circle.

Arc Length Formula

Once you've got your central angle in radians, the next step is to understand the arc length formula. The arc length \(s\) of a circle depends on the radius \(r\) and the central angle \(\theta\) in radians. The formula is \(s = r\theta\), a beautifully simple representation showing the direct relationship between the arc length, the radius of the circle, and the angle subtended at the center.

This formula is very powerful because it allows you to find the length of an arc, which is a portion of the circumference, by only knowing the circle's radius and the angle. If the central angle were to span the whole circle, \(\theta\) would be \(2\pi\) radians, and thus \(s\) would be equal to the circle's entire circumference, which is \(2\pi r\).

By recognizing this relationship, you're better equipped to solve circle-related problems. For example, with a central angle of \(\frac{3\pi}{4}\) radians and an arc length of 82 miles, you use the formula \(82 = r \times \frac{3\pi}{4}\) to find the radius.

Solving for Radius

After you've grasped the arc length formula and how to convert angles to radians, you're in a good position to solve for the radius. The radius of a circle is not only a measure of its size but also a key part of many other properties, including the area and the circumference. When you're given the arc length and the central angle, you can rework the arc length formula \(s = r\theta\) to solve for \(r\), making \(r = \frac{s}{\theta}\).

In the context of our example, where we have the arc length and the central angle in radians, we want to isolate \(r\) which gives us the equation \(r = \frac{82}{\frac{3\pi}{4}}\). Now, by multiplying both sides of the equation by the reciprocal of the fraction \(\frac{3\pi}{4}\), you can find the radius directly. This is basic algebra - multiplying by the reciprocal is like a 'cancelling out' operation. In practice, this looks like \(r = 82 \times \frac{4}{3\pi}\), and this computation will result in the radius measure of the circle.

Being comfortable with converting, substituting, and solving in this way allows you to tackle a variety of geometry problems related to circles, and understanding these steps deeply is crucial for success in geometry.