In a circle with center O, central angle AOB has a measure of \frac{5\pi}{6} radians. If t… — SAT

In a circle with center $O$ , central angle $AOB$ has a measure of $\frac{5\pi}{6}$ radians. If the radius of the circle is $12$ , the length of arc $AB$ is $k\pi$ . What is the value of $k$ ?

Answer: 10

Answer

The value of

k

10

The arc length

s

subtended by a central angle

\theta

(in radians) in a circle of radius

r

is given by

s = r\theta

. Substituting the radius

r = 12

and the angle

\theta = \frac{5\pi}{6}

yields

s = 12 \times \frac{5\pi}{6} = 10\pi

. Since the arc length is expressed as

k\pi

, the value of

k

10

Step-by-Step Solution

Identify the formula for arc length in radians.

s = r\theta

The arc length

s

is directly proportional to the radius

r

and the central angle

\theta

in radians.

Substitute the given values into the formula.

s = 12 \times \frac{5\pi}{6}

The radius of the circle is

12

and the central angle is

\frac{5\pi}{6}

radians.

Calculate the arc length.

s = 10\pi

Simplifying

12 \times \frac{5}{6}

gives

2 \times 5 = 10

, so the product is

10\pi

Determine the value of

k

k = 10

We equate the calculated arc length

10\pi

with the given expression

k\pi

Key Concept

Arc length of a circle using radian measure

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic