In a circle with center O, the length of minor arc AB is 3\pi and the area of sector AOB i… — SAT

In a circle with center $O$ , the length of minor arc $AB$ is $3\pi$ and the area of sector $AOB$ is $18\pi$ . What is the circumference of the circle?

A
$12\pi$
B
$18\pi$
$24\pi$ Cevap
D
$36\pi$

Cevap

The circumference of the circle is

24\pi

The correct answer is the option containing

24\pi

. By using the relationship

A = \frac{1}{2}rs

, where

A

is the sector area,

r

is the radius, and

s

is the arc length, we substitute the given values to get

18\pi = \frac{1}{2}r(3\pi)

. Solving for the radius yields

r = 12

. Substituting this radius into the circumference formula

C = 2\pi r

gives

2\pi(12) = 24\pi

Adım Adım Çözüm

Write the formulas for arc length

s

and sector area

A

in terms of radius

r

and central angle

\theta

in radians.

s = r\theta = 3\pi

and

A = \frac{1}{2}r^2\theta = 18\pi

This sets up the system of equations using the given geometric properties.

Express the sector area formula in terms of arc length by substituting

s = r\theta

into

A = \frac{1}{2}r(r\theta)

A = \frac{1}{2}rs

, which becomes

18\pi = \frac{1}{2}r(3\pi)

This simplifies the relationship to a single equation with one variable,

r

Solve the equation

18\pi = 1.5\pi r

for the radius

r

r = 12

Finding the radius is necessary to calculate the circumference of the circle.

Substitute

r = 12

into the circumference formula

C = 2\pi r

C = 2\pi(12) = 24\pi

This provides the final circumference value requested by the question.

Anahtar Kavram

Relationship between arc length, sector area, and circumference in circle geometry

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru