Problem 11 - Precalculus Essentials

Question

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius, \(r\) 1 meter Arc Length, \(s\) 400 centimeters

Step-by-Step Solution

Answer

The radian measure of the central angle that intercepts a 400 centimeter arc on a circle with a radius of 1 meter is 4 radians.

Step 1: Convert units

Since the radius \(r\) and the length of the arc \(s\) are not in the same units, we need to convert them to the same unit. Here, we convert the arc length in centimeters to meters. So, \(s = 400 cm = 4m \).

Step 2: Rearrange the formula

We need to find the radian measurement (\(\theta\)), which we can get by rearranging \(s = r* \theta\) to \(\theta = \frac{s}{r}\).

Step 3: Substitute the values into the new formula

Substitute the values into the new formula to find \(\theta\). So, \(\theta = \frac{s}{r} = \frac{4 m}{1 m} = 4 \) radians.

Step 4: Solution

Therefore, the radian measure of the central angle that intercepts the 400 centimeter arc on a circle with a radius of 1 meter is 4 radians.

Key Concepts

Understanding Central Angle

In the context of circles, a central angle is formed by two radii of the circle. It essentially divides the circle into two arcs. The central angle is located at the circle's center and measures how far the endpoints of the arc have been pulled apart. This angle can be measured in degrees or radians. When dealing with radian measures, the central angle is directly related to the arc length and radius. If the arc length equals the radius of the circle, the central angle is exactly one radian. Central angles play a crucial role in determining the size of the sector of a circle and relating arc lengths to circle radii.

Calculating Arc Length

Arc length is the distance along the curved line making up the arc. The relationship between the arc length, the radius of the circle, and the central angle is vital in circular geometry. The formula for finding the arc length \( s \) when the central angle \( \theta \) is given in radians is \( s = r \theta \). Here, \( r \) stands for the radius of the circle. This formula helps to directly connect how much of the circle's circumference a particular arc covers. If the central angle is in degrees, remember to convert it to radians before using it in this formula by employing unit conversion.

Importance of Unit Conversion

Unit conversion is essential when dealing with any mathematical problems involving measurements, especially in mixed unit scenarios. In the given problem, the radius is in meters, and the arc length is in centimeters. To streamline calculations and reduce errors, convert all measurements to the same unit. In this case, converting centimeters to meters provides a coherent unit measurement system. 100 centimeters equals 1 meter. Therefore, 400 cm becomes 4 meters, aligning with the radius's unit. This step ensures that calculations for the radian measure of the central angle remain accurate and comprehensible.

Determining Circle Radius

The radius of a circle is a fundamental element in circular geometric calculations. It is the distance from the center of the circle to any point on its circumference. The radius is crucial when calculating the central angle, arc length, or even the circumference and area of the circle. Having the correct unit for the radius is vital when performing any calculations related to the circle. In problems like this exercise, ensure that all measurements are consistent to allow seamless application of formulas. The radius plays a central role in converting arc measurements into angle measurements and vice versa, showcasing its importance in understanding and solving circular geometry problems.