A circular swimming pool cover has a total area of 64\pi square feet. A specific section o… — ACT

A circular swimming pool cover has a total area of $64\pi$ square feet. A specific section of the cover, formed by a circular sector with a central angle of $135^\circ$ , is made of a reinforced heavy-duty material. What is the perimeter, in feet, of this reinforced sector?

A
$6\pi$
$6\pi + 16$ Cevap
C
$6\pi + 8$
D
$12\pi + 16$
E
$24\pi + 16$

Cevap

The perimeter of the reinforced sector is

6\pi + 16

feet.

First, find the radius from the circle area formula:

\pi r^2 = 64\pi \implies r = 8

feet. Next, calculate the arc length of the sector using the angle fraction

\frac{135^\circ}{360^\circ} = \frac{3}{8}

. Multiplying by the total circumference

2\pi(8) = 16\pi

gives an arc length of

6\pi

feet. Finally, add the two straight radius edges to get the total perimeter:

6\pi + 8 + 8 = 6\pi + 16

feet.

Adım Adım Çözüm

Find the radius of the circular pool cover using the given area.

Since Total Area

= \pi r^2 = 64\pi

, dividing by

\pi

yields

r^2 = 64

, so

r = 8

feet.

The radius is required to determine both the arc length and the lengths of the straight boundary edges.

Calculate the arc length of the sector.

Arc length

= \frac{\theta}{360^\circ} \times 2\pi r = \frac{135^\circ}{360^\circ} \times 2\pi(8) = \frac{3}{8} \times 16\pi = 6\pi

feet.

The curved boundary of the sector is a fraction of the circle's full circumference.

Calculate the total perimeter of the sector by adding the arc length and the two radii.

Perimeter

= \text{Arc length} + 2r = 6\pi + 2(8) = 6\pi + 16

feet.

The perimeter of any sector consists of its curved outer arc plus its two straight side radii.

Anahtar Kavram

Perimeter of a Circular Sector

Tahmini Süre:1m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru