In circle O, sector AOB has an area of 45\pi square units and arc AB has a length of 6\pi … — GRE General Test

In circle $O$ , sector $AOB$ has an area of $45\pi$ square units and arc $AB$ has a length of $6\pi$ units. What is the perimeter of sector $AOB$ ?

A
$15 + 6\pi$
$30 + 6\pi$ Cevap
C
$45 + 6\pi$
D
$60 + 6\pi$
E
$30\pi$

Cevap

30 + 6\pi

The area of a sector is given by

\frac{1}{2} r L

, where

r

is the radius and

L

is the arc length. Substituting

L = 6\pi

and Area

= 45\pi

yields

45\pi = \frac{1}{2} r (6\pi)

, which simplifies to

45\pi = 3\pi r

, so

r = 15

. The perimeter of the sector is

L + 2r = 6\pi + 2(15) = 30 + 6\pi

. Thus, the option equal to

30 + 6\pi

is correct.

Adım Adım Çözüm

Express sector area and arc length in terms of radius

r

and central angle

\theta

in degrees.

Sector Area =

\frac{\theta}{360^\circ} \times \pi r^2 = 45\pi

and Arc Length =

\frac{\theta}{360^\circ} \times 2\pi r = 6\pi

Relating both formulas allows solving for the radius directly using the ratio of area to arc length.

Divide the sector area formula by the arc length formula to find radius

r

\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{\frac{\theta}{360^\circ} \pi r^2}{\frac{\theta}{360^\circ} 2\pi r} = \frac{r}{2} = \frac{45\pi}{6\pi} = 7.5 \implies r = 15.

The central angle fraction

\frac{\theta}{360^\circ}

and

\pi

cancel out, leaving

\frac{r}{2} = 7.5

Calculate the total perimeter of sector

AOB

\text{Perimeter} = \text{Arc Length} + 2r = 6\pi + 2(15) = 30 + 6\pi.

The perimeter of a sector consists of the outer arc length plus the two straight boundary radii (

OA

and

OB

Anahtar Kavram

The relationship between sector area, arc length, radius, and sector perimeter

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Konular

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