A garden bed is designed in the shape of a sector of a circle with a central angle measuri… — ACT

A garden bed is designed in the shape of a sector of a circle with a central angle measuring $120^\circ$ . If the total perimeter of the sector-shaped garden bed is $24 + 8\pi$ feet, what is the area of the garden bed, in square feet?

A
$8\pi$
B
$24\pi$
$48\pi$ Cevap
D
$96\pi$
E
$144\pi$

Cevap

The area of the garden bed is

48\pi

square feet.

The perimeter of a sector is given by

2r + \frac{\theta}{360^\circ}(2\pi r)

. Substituting

\theta = 120^\circ

gives

2r + \frac{2\pi r}{3} = 24 + 8\pi

, which simplifies to

r = 12

feet. Using the sector area formula

\frac{\theta}{360^\circ}\pi r^2

with

r = 12

yields

\frac{1}{3}\pi(144) = 48\pi

square feet.

Adım Adım Çözüm

Set up the formula for the perimeter of a sector.

Perimeter

= 2r + \text{Arc Length} = 2r + \left(\frac{\theta}{360^\circ}\right)(2\pi r)

A sector's perimeter consists of two straight radii and the curved arc boundary.

Substitute

\theta = 120^\circ

into the perimeter expression and solve for the radius

r

2r + \left(\frac{120^\circ}{360^\circ}\right)(2\pi r) = 24 + 8\pi \implies 2r + \frac{2\pi r}{3} = 24 + 8\pi \implies r\left(2 + \frac{2\pi}{3}\right) = 12\left(2 + \frac{2\pi}{3}\right) \implies r = 12

feet

Equating the algebraic expression to the given perimeter allows finding the circle radius.

Calculate the sector area using

r = 12

feet and

\theta = 120^\circ

Area

= \left(\frac{120^\circ}{360^\circ}\right) \pi r^2 = \frac{1}{3} \pi (12)^2 = \frac{144\pi}{3} = 48\pi

square feet

The area of a sector is the fraction of the total circle's area defined by the ratio of central angle to

360^\circ

Anahtar Kavram

Perimeter and Area of a Circle Sector

Tahmini Süre:1m 0s

Soru

Cevap

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