Select the correct answer.
A circle has a radius of 22 centimeters. Arc XY has a length of $\frac{66}{5} \pi$ centimeters. What is the radian measure of the corresponding central angle?
A. $\frac{2}{3} \pi$
B. $\frac{3}{5} \pi$
C. $\frac{4}{5} \pi$
D. $\frac{2}{5} \pi
Answer (2)
We are given the radius and arc length of a circle.
We use the formula s = r θ to relate arc length, radius, and central angle.
We substitute the given values into the formula and solve for the central angle θ .
The radian measure of the central angle is 5 3 π .
Explanation
Problem Analysis We are given a circle with radius r = 22 cm and an arc XY with length s = 5 66 π cm. We need to find the radian measure θ of the central angle corresponding to the arc XY.
Formula Introduction The formula relating arc length, radius, and central angle in radians is:
s = r θ
where:
s is the arc length,
r is the radius,
θ is the central angle in radians.
Substitution Substitute the given values of s and r into the formula:
5 66 π = 22 θ
Solving for theta Solve for θ :
θ = 22 5 66 π = 5 ⋅ 22 66 π = 5 3 π
Final Answer The radian measure of the central angle is 5 3 π .
Examples
Imagine you're designing a pizza, and you want to cut a slice such that the arc length of the crust is a specific length. Knowing the radius of the pizza and the desired arc length, you can calculate the central angle of the slice using the formula s = r θ . This helps you cut the perfect slice every time!
The radian measure of the central angle corresponding to arc XY is 5 3 π . This was calculated using the arc length formula s = r θ with the given values. The final answer is option B.
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