I don't understand what steps I have to do in order to solve this problem.
An angle's rays cut off an arc length of 227 centimeters in a circle with a 40 centimeter radius.
A. What portion of the circle's circumference is cut off by this angle?
B. What is the measure of this angle in radians?
C. What is the measure of this angle in degrees?
the entire circumference is 40*2π = 251.33 cm
so, 227/251.33 = 0.9 of the whole circumference
the angle is thus 0.9 * 2π = 1.8π radians
1.8π * 180/π = 324°
Old sorry
To solve each part of this problem, we need to understand some key concepts related to circles and angles. Let's break down each part step by step:
A. To determine the portion of the circle's circumference cut off by the angle, we need to find the ratio of the length of the cut-off arc to the circumference of the entire circle. Here's the step-by-step process:
1. Calculate the circumference of the circle using the formula: Circumference = 2 * π * radius. In this case, the radius is given as 40 centimeters, so the formula becomes: Circumference = 2 * 3.14 * 40 = 251.2 centimeters.
2. Take the ratio of the length of the cut-off arc to the circumference of the entire circle. The cut-off arc length is given as 227 centimeters, so the ratio is: (Cut-off Arc Length / Circumference) * 100.
(227 / 251.2) * 100 = 90.42%.
Therefore, the angle cuts off approximately 90.42% of the circle's circumference.
B. To calculate the measure of the angle in radians, we can use the formula: Angle (in radians) = Arc Length / Radius. We already know the arc length (227 centimeters) and the radius (40 centimeters) from the problem:
Angle (in radians) = 227 / 40 = 5.68 radians.
Therefore, the measure of the angle is approximately 5.68 radians.
C. To convert the measure of the angle from radians to degrees, we need to use the conversion formula: Angle (in degrees) = Angle (in radians) * (180 / π). Using the angle we found in part B (5.68 radians):
Angle (in degrees) = 5.68 * (180 / 3.14) = 325.47 degrees.
Therefore, the measure of the angle is approximately 325.47 degrees.
By following these step-by-step explanations, you'll be able to solve this problem and understand how to approach similar problems in the future.