Which of the following quantities are affected by doubling the lengths of the chains: the radius of the arc or the measure of the angle?
A
a
B
b
11Ï€/12 radians
C
c
18.7 feet
D
d
Answer: The arc length would double.
Explanation: See solution.
Practice makes perfect
a
A swing has a 165^(∘) angle of rotation. We are asked to draw the angle in standard position. To do that, we first need to locate the initial side of the angle on the positive x-axis.
Having done that, we are going to draw the terminal side of the 165^(∘) angle counterclockwise past the positive x-axis.
b
We need to write the degree measure of 165^(∘) in radians. To do that, we will multiply the measure by a conversion factor of π radians180^(∘) and simplify.
The chains of a swing are 6 12 feet long, which means that the swing has a radius of 6 12 = 6.5 feet. We need to find the length of an arc formed by the swing.
Let's recall the formula for the length of an arc s. In the formula r is the radius and θ is the measure of the angle of rotation, expressed in radians.
s=rθHere we are given the radius but we are missing the angle. Notice that we can use the radian measure of the angle of rotation found in Part B.
165^(∘)= 11π/12
Let's substitute the radian measure, as well as the radius of the arc, into the formula for the arc length and simplify.
We are asked to describe how the arc length would change if the lengths of the chains of the swing were doubled. To do that, let's take a closer look at the formula for the arc length.
s= r θ
The length of an arc depends on two quantities.
The radius of the arc
The angle of rotation.
Doubling the length of the chains only affects the radius of the arc r. The angle of rotation θ remains unchanged. Since in the formula r is not raised to any power, if r is doubled so is the arc length.
