Problem 36 - Algebra 2 and Trigonometry

Question

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(s=18\) and \(\theta=\frac{6 \pi}{5},\) find \(r\)

Step-by-Step Solution

Answer

The radius \( r \) is \( \frac{15}{\pi} \).

Step 1: Understand the Formula

We know that the formula for the length of an arc when the central angle is given in radians is \( s = r \theta \). Here, \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.

Step 2: Substitute the Given Values

We have been given \( s = 18 \) and \( \theta = \frac{6 \pi}{5} \). Substitute these values into the formula: \( 18 = r \times \frac{6\pi}{5} \).

Step 3: Solve for the Radius

To solve for \( r \), we need to isolate \( r \) in the equation \( 18 = r \times \frac{6\pi}{5} \). Divide both sides by \( \frac{6\pi}{5} \): \[ r = \frac{18}{\frac{6\pi}{5}} \].

Step 4: Simplify the Expression

Simplify the fraction \( \frac{18}{\frac{6\pi}{5}} \) by multiplying by the reciprocal of \( \frac{6\pi}{5} \), which is \( \frac{5}{6\pi} \). Thus, \( r = 18 \times \frac{5}{6\pi} \).

Step 5: Calculate the Exact Value

Multiply \( 18 \times \frac{5}{6\pi} \) to find \( r \), which gives \( r = \frac{18 \times 5}{6\pi} = \frac{90}{6\pi} = \frac{15}{\pi} \).

Step 6: Result Interpretation

Thus, the radius \( r \) of the circle is \( \frac{15}{\pi} \).

Key Concepts

Arc Length Formula

The arc length formula is crucial in understanding the geometry of a circle. It connects the concepts of arc length, radius, and central angle. When working with radians, the formula is expressed as \( s = r \theta \). Here:

• \( s \) is the arc length
• \( r \) is the circle's radius
• \( \theta \) is the central angle in radians

This formula allows us to easily find one of these values if we have the other two. For example, knowing the arc length and the central angle, we can calculate the radius. Conversely, with the radius and central angle, the arc length can be determined. This formula is derived from the proportion of the arc length to the circumference of a circle, which is \( 2\pi r \). Understanding this concept provides the foundation for calculating other geometric measurements in circles.

Central Angle

The central angle is an angle whose vertex is at the center of the circle and whose sides intersect the circle, creating an arc. It plays a significant role in determining the arc's length. In the context of radians, the measure of a central angle is directly tied to the length of the arc it intercepts by the formula \( s = r \theta \).
Central angles can be expressed in degrees or radians, but radians provide a more natural measure connected to arc length. Unlike degrees, which divide a circle into 360 parts, radians measure based on the circle's radius.

• One complete revolution around a circle is \( 2\pi \) radians.
• A semicircle is \( \pi \) radians.
• A right angle equates to \( \frac{\pi}{2} \) radians.

Understanding central angles in radians simplifies calculations involving arc length and allows for more seamless conversions between different circular measurements.

Radius Calculation

Calculating the radius of a circle, especially when given the arc length and central angle, is a straightforward process using the arc length formula. Suppose you know the arc length \( s \) and the central angle \( \theta \), the formula \( s = r \theta \) can be rearranged to solve for the radius \( r \). This involves isolating \( r \) through the equation transformation:\[ r = \frac{s}{\theta} \]This process allows you to understand the size of the circle given how much of the circumference the arc covers and the size of the angle. Let's consider an example:
If the arc length \( s \) is 18 and the angle \( \theta \) is \( \frac{6\pi}{5} \), substituting into the formula gives:
\[ r = \frac{18}{\frac{6\pi}{5}} \] which simplifies further to \( r = \frac{15}{\pi} \). Thus, knowing two of these core elements enables you to determine the third, emphasizing the simplicity and efficiency of using radians in circular calculations.

Radians to Arc Length Conversion

Converting radians to arc length involves understanding the relationship between the angle, the radius, and the arc length itself. This process is encapsulated in the formula \( s = r \theta \), where extending any radian measure by multiplying it by the radius of the circle gives the precise arc length.
When you want to find the length of an arc, simply multiply the central angle \( \theta \) (in radians) by the radius \( r \):

• If \( r = \frac{15}{\pi} \) and \( \theta = \frac{6\pi}{5} \), the multiplication leads to the arc length \( s = \left(\frac{15}{\pi}\right) \times \left(\frac{6\pi}{5}\right) = 18 \).

This conversion highlights the elegance of working with radian measures, making it easy to transform angular measures into actual lengths on the circle's circumference. This not only simplifies mathematical calculations but also clarifies geometric interpretations.