An arc on a circle measures $85^{\circ}$. The measure of the central angle, in radians, is within which range?
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 $\pi$ radians
Answer (2)
Convert the angle from degrees to radians using the formula: radians = degrees × 18 0 ∘ π .
Calculate the radian measure: 8 5 ∘ × 18 0 ∘ π = 36 17 π ≈ 1.48 radians.
Compare the calculated radian measure to the given ranges.
Determine that the radian measure falls within the range: 0 to 2 π radians. The final answer is 0 to 2 π radians .
Explanation
Problem Analysis The problem provides the measure of an arc in degrees and asks us to find the range in which the corresponding central angle in radians lies. To solve this, we need to convert the angle from degrees to radians and then compare the result with the given ranges.
Conversion Formula To convert an angle from degrees to radians, we use the conversion factor 18 0 ∘ π . So, we multiply the angle in degrees by this factor to get the equivalent angle in radians. Angle in radians = Angle in degrees × 18 0 ∘ π
Conversion Calculation Now, let's convert 8 5 ∘ to radians: 8 5 ∘ × 18 0 ∘ π = 180 85 π = 36 17 π radians The result of the operation is approximately 1.48 radians.
Range Comparison We need to determine which of the given ranges contains 1.48 radians.
0 to 2 π radians: 2 π ≈ 1.57 radians. Since 0 < 1.48 < 1.57 , this range is a possibility.
2 π to π radians: π ≈ 3.14 radians. Since 1.57 < 1.48 < 3.14 is false, this range is not correct.
π to 2 3 π radians: This range starts at approximately 3.14 radians, so it's not correct.
2 3 π to 2 π radians: This range is also not correct since it starts at a value greater than 1.48.
Final Answer Since 1.48 is between 0 and 2 π ≈ 1.57 , the measure of the central angle in radians is within the range of 0 to 2 π radians.
Examples
Understanding how to convert between degrees and radians is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a pendulum, you need to work with angles in radians to apply the equations of motion correctly. Similarly, in computer graphics, angles are often represented in radians when performing rotations and transformations of objects on the screen. Knowing the radian measure helps in precise calculations and simulations in these real-world applications.
The central angle corresponding to an arc measuring 8 5 ∘ converts to approximately 1.48 radians. This value falls within the range of 0 to 2 π radians. Therefore, the correct answer is option A.
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