Question

s denotes the length of the arc of a circle of radius \(r\) subtended by the central angle \(\theta .\) Find the missing quantity. Round answers to three decimal places. \(\theta=\frac{2}{3}\) radian, \(s=8\) feet, \(r=?\)

Short answer

The radius \(r\) is 12 feet.

Step-by-step solution

Write down the formula for arc length

The formula to find the arc length of a circle is given by \[ s = r \theta \]where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.

Substitute the known values

Insert the given values into the formula. The values provided are \(\theta = \frac{2}{3}\) radians and \(s = 8\) feet. Substitute them into the formula:\[ 8 = r \left( \frac{2}{3} \right) \]

Solve for the radius \(r\)

Rearrange the equation to solve for \(r\). Multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \):\[ r = \frac{8}{ \frac{2}{3} } \]Simplify the fraction by multiplying the numerator by the reciprocal of the denominator:\[ r = 8 \times \frac{3}{2} \]\[ r = 12 \]

Round to the desired decimal places

The value for \(r\) is already an integer, so no rounding is necessary. Thus the radius \(r\) is 12 feet.

Key concepts

Arc Length Formula

The arc length formula is essential when dealing with segments of a circle. This formula is written as \( s = r \theta \), where \( s \) stands for the arc length, \( r \) is the radius of the circle, and \( \theta \) represents the central angle in radians. By knowing this formula, you can determine the length of an arc when you have the radius and central angle. It's a straightforward multiplication, making it easy to find one variable if you know the other two. This formula connects the geometry of a circle directly to its properties in a very practical way.

Central Angle in Radians

A crucial part of using the arc length formula is understanding the central angle measured in radians. Radians are units for measuring angles that relate directly to the radius of the circle. One full revolution around a circle is \(2π\) radians. Therefore, \( \theta \) in the formula \( s = r \theta \) must be in radians for the calculation to be correct. To convert degrees to radians, use the conversion factor: \(1 \text{ degree} = \frac{\text{π}}{180} \text{ radians}\). This concept helps bridge the gap between angular measurements and real-world distances on the circle's circumference.

Circle Radius Calculation

Circle radius calculation comes into play heavily with the arc length formula. To find the radius \( r \) of a circle when given the arc length \( s \) and the central angle \( \theta \), rearrange the formula \( s = r \theta \). This rearrangement looks like \( r = \frac{s}{\theta} \). By plugging in the values for \( s \) and \( \theta \), you can solve for \( r \). For instance, using the example where \( s = 8 \) feet and \( \theta = \frac{2}{3} \) radians, the radius is calculated as follows: \( r = \frac{8}{\frac{2}{3}} = 12 \) feet. This process shows how the radius can be found with straightforward algebraic manipulation.