Question

How do you convert $$16$$ revolutions to radians?

Answer 1

Hint: In this given problem, we need to convert $$16$$ revolutions to radians. Here, we use the arc length formula to find the radians, $$s=r\theta$$ .The radians and evolutions are both units used to measure angle. A revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side.

Complete step by step solution:
In the given problem,
Here, $$16$$ revolutions to radians
We need to convert the given revolution to radians, we get
We know that,
Radians of $$2\pi$$ becomes one revolution
By apply arc length formula is given below,
 $$s=r\theta$$
Let the angle, $$\theta =2\pi$$ , which is the formula for the full circumference of a circle.
Since, $$2\pi$$ radians is one revolution.
 $$\Rightarrow s=r\theta$$ ,
Where $$r=16rev$$ and angle, $$\theta ={\displaystyle \frac{2\pi rad}{1rev}}$$ .
By substituting the values into the arc length formula.
 $$s=16rev\left({\displaystyle \frac{2\pi rad}{1rev}}\right)$$
Where, $$rev$$ denotes revolution and $$rad$$ means radians.
By simplify it in further step, we get
 $$s=32\pi rad$$ . Since, radians of $$2\pi$$ becomes one revolution
Therefore, the conversion of $$16$$ revolutions to radians is $$32\pi rad$$
So, the correct answer is “ $$32\pi rad$$ ”.

Note: We note that one revolution is $$2\pi$$ radians and a radian is the angle subtended by an arc of length equal to the radius. That is, if the radius is $$r$$ then the length of the arc is $$r$$ . For an arc to subtend a full revolution its length must be 2πr so the angle is 2π radians. Radians are used to measure angles. You might be more used to measuring angles with degrees, in which case it should help to think of radians as a different sized unit to measure the same thing. A $${360}^{\circ }$$ degree angle is the same as a $$2\pi$$ radian angle.