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. \(r=10\) miles, \(s=9\) miles, \(\theta=?\)

Short answer

The central angle \(\theta\) is 0.9 radians.

Step-by-step solution

Write Down the Formula

The formula relating the arc length, radius, and central angle is given by: \[ s = r \theta \]

Substitute Given Values

Substitute the given values for arc length (s) and radius (r) into the formula: \[ 9 = 10 \theta \]

Solve for \(\theta\)

To find the value of the central angle \(\theta\), divide both sides of the equation by 10: \[ \theta = \frac{9}{10} \]

Simplify and Round

Simplify the fraction to get: \[ \theta = 0.9 \] Since the question asks to round the answer to three decimal places, we confirm: \[ \theta = 0.900 \]

Key concepts

Central Angle

A central angle is the angle formed when two radii meet at the center of a circle. It plays a crucial role in understanding various properties of a circle, especially when it comes to measuring arc length. This angle is measured in radians when dealing with arc length problems. To get a better grasp, imagine the hands of a clock forming an angle at the center as they point towards different numbers on the clock. The space between these hands represents the central angle.

Circle Radius

The circle radius is the distance from the center of the circle to any point on its circumference. It is a fundamental property of circles and is half of the diameter. Knowing the radius is vital for calculating various measurements, such as the circumference and the area of a circle. For example, if you know the radius and the central angle, you can find the arc length of the circle, which is directly proportional to the radius.

Arc Length Formula

The formula to calculate the arc length of a circle is: s = rθ where: s is the arc length, r is the radius, θ is the central angle in radians. Understanding this formula is essential, as it links the three key quantities together. In this exercise, we are given the radius and the arc length, and we need to find the central angle.

Solving Equations

Solving equations is a fundamental skill in mathematics, necessary for finding unknown values. In the context of our problem, we use the formula for arc length and manipulate it to find the unknown central angle, θ. First, we substituted the known values into the formula: s = rθ 9 = 10θ Then, we isolate θ by dividing both sides by the circle radius (10): θ = 9 ÷ 10 Simplifying this, we get θ = 0.9. Finally, rounding to three decimal places, the central angle, θ, is 0.900 radians. Solving equations often involves these similar steps: substitute known values, isolate the unknown variable, and simplify.