Problem 140 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). $$r=15 \text { in. }, \theta-110^{\circ}$$

Step-by-Step Solution

Answer

The length of the arc is \( s = 15 \cdot \frac{110}{180} \pi \) inches.

Step 1: Convert Degrees into Radians

To convert the angle from degrees to radians, use the formula \( radians = \frac{degrees}{180} \pi \). Thus, \( \theta = \frac{110}{180} \pi \) radians.

Step 2: Substitute Into Arc Length Formula

Now that we have the angle in radians, we can substitute our given values into the arc length formula. This means \( s = 15 \cdot \frac{110}{180} \pi \).

Step 3: Simplify the Result

To find the length of the arc, simplify the expression. Multiply \( 15 \) by \( \frac{110}{180} \pi \) to get the arc length, \( s \).

Key Concepts

Converting Degrees to Radians

When working with circle measures, converting from degrees to radians is a crucial step. To do this, we use the fact that 180 degrees is equivalent to \( \pi \) radians. The conversion formula is \( radians = \frac{degrees}{180} \pi \).

For example, if you have an angle of 110 degrees, as in our exercise, you convert it to radians by dividing 110 by 180 and then multiplying the result by \( \pi \) to get \( \frac{110}{180} \pi \). Understanding this conversion is essential because many mathematical formulas, including those for finding arc lengths, require angles to be in radians.

Central Angles

Central angles have their vertex at the center of the circle and the sides are radii of the circle. The measure of a central angle tells us a lot about the circle it's within, especially when we're dealing with arc lengths. If you imagine stretching out the arc to form a straight line, the length of this line is the arc length.

The size of the central angle, measured in radians, is directly proportional to the length of the arc it intercepts. This means a larger central angle will correspond to a longer arc on the circle. Hence, for our exercise with a central angle of 110 degrees (which we converted to radians), it cuts out a significant portion of the circle's circumference.

Arc Length Formula

The arc length of a circle can be found by using the arc length formula, which is \( s = r\theta \), where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

Using this formula requires that the angle is in radians, not degrees, which is why conversion is our first crucial step. In the given example, after converting 110 degrees to radians, we substitute the radius (15 inches) and the angle in radians into the formula to calculate the arc length. Multiply the radius by the angle to find the length of the arc, which completes the process of finding how long the arc is based on the central angle and the size of the circle.