Question

Skytrek Adventure Park in Revelstoke, British Columbia, has a sky swing. Can you imagine a \(170 -ft\) flight that takes riders through a scary pendulum swing? At one point you are soaring less than \(10 \mathrm{ft}\) from the ground at speeds exceeding \(60 \mathrm{mph}\). a) The length of the cable is 72 ft and you travel on an arc of length \(170 \mathrm{ft}\) on one particular swing. What is the measure of the central angle? Give your answer in radians, to the nearest hundredth. b) What is the measure of the central angle from part a), to the nearest tenth of a degree?

Short answer

a) 2.36 radians. b) 135.2 degrees.

Step-by-step solution

Understand the Given Values

Identify the given values: the length of the cable (radius) is 72 ft and the arc length is 170 ft. We need to find the measure of the central angle.

Use the Formula to Find the Central Angle in Radians

The formula to find the central angle in radians is \[ \theta = \frac{s}{r} \] where \(s\) is the arc length and \(r\) is the radius. Substituting the given values into the formula: \[ \theta = \frac{170}{72} \approx 2.36 \] So, the central angle in radians, to the nearest hundredth, is 2.36 radians.

Convert the Central Angle to Degrees

To convert radians to degrees, use the formula \[ \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \] Substituting \( \theta = 2.36 \), \[ \theta_{degrees} = 2.36 \times \frac{180}{\pi} \approx 135.2 \] So the central angle, to the nearest tenth of a degree, is 135.2 degrees.

Key concepts

Arc Length

Arc length is a portion of the circumference of a circle. It represents the distance between two points along the circle's edge. In the context of a sky swing ride, the arc length tells us how much of the swing's path is along a circular trajectory. To find the arc length, use the formula \[ s = r \theta \] where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. If you know the arc length and radius, you can rearrange the formula to find the central angle: \[ \theta = \frac{s}{r} \] Understanding this concept is crucial for solving problems involving circular motion and angles.

Radian Measure

The radian is a unit of angular measure used in many areas of mathematics. One radian is the angle created when the arc length is equal to the radius of the circle. Thus, if you travel a distance along the circle equal to the radius, you've moved through an angle of 1 radian. Radians are particularly useful because they simplify the relationship between arc length, radius, and angle. For example, in the given problem, we are using the formula \[ \theta = \frac{s}{r} \] to find the central angle in radians from the arc length and radius. When solving circular motion problems, working with radians often makes the math easier.

Degree Conversion

Degrees are another way to measure angles, commonly used in everyday life. There are 360 degrees (°) in a full circle, while there are approximately 6.283 (or 2\(\pi\)) radians in a full circle. To convert between radians and degrees, use the following formulas:

• From radians to degrees: \[ \theta_{degrees} = \theta_{radians} \times \frac{180}{\tau} \]
• From degrees to radians: \[ \theta_{radians} = \theta_{degrees} \times \frac{\tau}{180} \]

In the problem, we converted the central angle from radians to degrees by using \[ \theta_{degrees} = 2.36 \times \frac{180}{\tau} \] to find an angle of 135.2 degrees. Understanding these conversions is essential in many trigonometric and geometry problems.

Trigonometric Formulas

Trigonometry often deals with relationships involving angles and distances in circles. Useful trigonometric formulas include:

• \( \theta = \frac{s}{r} \): Relates arc length, radius, and central angle.
• Sine and Cosine functions: \[ \text{For a right triangle,} \] \[ \text{Sin}(\theta) = \frac{opposite}{hypotenuse}, \] \[ \text{Cos}(\theta) = \frac{adjacent}{hypotenuse} \] or in a unit circle.
• Pythagorean identity: \[ \text{Sin}^2(\theta) + \text{Cos}^2(\theta) = 1 \]

These relationships help solve various problems involving circular and angular motion. For our problem, while we didn't use sine or cosine directly, understanding the geometry of circles and their angles underpins the calculations.

Circular Motion

Circular motion describes the movement of an object along the circumference of a circle. It can be uniform, where the speed is constant, or non-uniform, where the speed varies. Essential parameters in circular motion include the radius, speed, and central angle. In our given problem, the sky swing follows a circular path, with the arc length representing part of this path. Central angle calculation is a crucial aspect of understanding the object's position and speed at different points. The formulas and concepts discussed help describe and predict the motion accurately. By understanding these core ideas, you can better grasp problems involving circles and motion, especially those that involve transitioning between different units of angular measure.