Rob and his brother are riding in the same car on a Ferris wheel which has a diameter of 40 feet. Once the Ferris wheel starts, they travel a distance of [tex]$\frac{86}{3} \pi$[/tex] feet before the ride stops to let more people on. What is the measure of the associated central angle for the arc they traveled?
The central angle measures $\square$ ${ }^{\circ}$.
Answer (2)
Calculate the circumference of the Ferris wheel: C = 2 π r = 40 π feet.
Determine the fraction of the circumference traveled: 40 π 3 86 π = 60 43 .
Multiply the fraction by 36 0 ∘ to find the central angle: 60 43 × 36 0 ∘ = 25 8 ∘ .
The measure of the central angle is: 258
Explanation
Problem Analysis First, let's analyze the problem. We are given the diameter of a Ferris wheel, which is 40 feet. This means the radius is half of that, which is 20 feet. We are also given the distance traveled by Rob and his brother, which is 3 86 π feet. Our goal is to find the measure of the central angle associated with this arc length.
Calculate the Circumference Next, we need to find the circumference of the Ferris wheel. The formula for the circumference of a circle is C = 2 π r , where r is the radius. In this case, the radius is 20 feet, so the circumference is: C = 2 π ( 20 ) = 40 π feet
Find the Fraction of the Circumference Now, we need to determine what fraction of the total circumference the distance traveled represents. To do this, we divide the distance traveled by the circumference: circumference distance traveled = 40 π 3 86 π = 3 86 π ⋅ 40 π 1 = 3 × 40 86 = 120 86 = 60 43
Calculate the Central Angle Finally, to find the measure of the central angle in degrees, we multiply this fraction by 36 0 ∘ :
Central Angle = 60 43 × 36 0 ∘ = 43 × 60 360 = 43 × 6 = 25 8 ∘
Final Answer Therefore, the measure of the associated central angle for the arc they traveled is 25 8 ∘ .
Examples
Imagine you're designing a rotating sprinkler system for a circular garden. Knowing the diameter of the garden and how far you want the water to reach (the arc length), you can calculate the central angle the sprinkler needs to cover to efficiently water that section of the garden. This ensures no water is wasted and the entire desired area is irrigated evenly.
To calculate the central angle associated with the arc length traveled on the Ferris wheel, we first found the circumference to be 40 π feet. Then, we determined the fraction of the circumference traveled and multiplied it by 360 degrees, resulting in a central angle of 25 8 ∘ .
;
