An arc on a circle measures $250^{\circ}$. Within which range is the radian measure of the central angle?
A. 0 to $\frac{\pi}{2}$ radians
B. $\frac{\pi}{2}$ to $\pi$ radians
C. $\pi$ to $\frac{3 \pi}{2}$ radians
D. $\frac{3 \pi}{2}$ to $2 \pi$ radians
Answer (1)
Convert the degree measure to radians using the conversion factor 18 0 ∘ π .
Calculate the radian measure: 25 0 ∘ × 18 0 ∘ π = 18 25 π .
Approximate π ≈ 3.14 to get 18 25 π ≈ 4.36 .
Compare the calculated radian measure with the given ranges to determine the correct range: π to 2 3 π radians .
Explanation
Problem Analysis We are given an arc on a circle that measures 25 0 ∘ . We need to find the range in which the radian measure of the central angle lies. The ranges are:
0 to 2 π radians 2 π to π radians π to 2 3 π radians 2 3 π to 2 π radians
Convert Degrees to Radians To convert from degrees to radians, we use the conversion factor 18 0 ∘ π . So, we multiply the degree measure by this factor to get the radian measure: 25 0 ∘ × 18 0 ∘ π = 180 250 π = 18 25 π Now, let's approximate this value to determine the range.
Approximate and Compare We can approximate π ≈ 3.14 . Therefore, 18 25 π ≈ 18 25 × 3.14 ≈ 18 78.5 ≈ 4.36 Now let's check the ranges: 0 to 2 π ≈ 2 3.14 ≈ 1.57 2 π to π ≈ 3.14 π to 2 3 π ≈ 2 3 × 3.14 ≈ 4.71 2 3 π to 2 π ≈ 2 × 3.14 ≈ 6.28 Since 4.36 is between 3.14 and 4.71 , the radian measure lies in the range π to 2 3 π radians.
Final Answer The radian measure of the central angle is 18 25 π , which is approximately 4.36 radians. This value falls within the range of π to 2 3 π radians.
Examples
Understanding radian measures is crucial in fields like physics and engineering, especially when dealing with circular motion or wave phenomena. For instance, when calculating the angular velocity of a rotating object, expressing angles in radians simplifies the equations and provides a more intuitive understanding of the object's speed and position. This conversion is also vital in signal processing, where frequencies are often expressed in radians per second to analyze and manipulate signals effectively.
