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In a circle with a radius of 7 feet, the radian measure of the central angle subtended by an arc with a length of 4 feet is ? The area of the sector formed by the arc is ? square feet.


Assume π = 3.14, and round your answers to the nearest hundredth.

Verified Answer :

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Part 1)

we know that

The circumference of a circle is equal to

\(C=2\pi r\)

where

r is the radius of the circle

In this problem we have

\(r=7\ ft\)

substitute

\(C=2\pi 7=14\pi\ ft\)

Remember that

\(2\pi\) radians subtends the complete circle of length \(14\pi\ ft\)

so

by proportion

Find the radian measure of the central angle subtended by an arc with a length of \(4\) feet

\(\frac{2\pi}{14\pi} \frac{radians}{feet} =\frac{x}{4} \frac{radians}{feet} \\ \\14\pi*x=4*2\pi \\ \\x=8/14 \\ \\ x=0.57\ radians\)

therefore

the answer Part 1) is

\(0.57\ radians\)

Part 2)

we know that

The area of a circle is equal to

\(A=\pi r^{2}\)

where

r is the radius of the circle

In this problem we have

\(r=7\ ft\)

substitute

\(A=\pi 7^{2}=49\pi\ ft^{2}\)

Remember that

The area of complete circle \(49\pi\ ft^{2}\) subtends a length of the complete circle of  \(14\pi\ ft\)

so

by proportion

Find the area of the sector formed by an arc with a length of \(4\) feet

\(\frac{49\pi}{14\pi} \frac{ft^{2}}{ft} =\frac{x}{4} \frac{ft^{2}}{ft} \\ \\14*x=49*4 \\ \\x=14 \ ft^{2}\)

therefore

the answer Part 2) is

\(14 \ ft^{2}\)

The area of the sector formed by the arc is 14 sq. ft.

What is the circumference of a circle?

The circumference of a circle is equal to 2πr.

where

r is the radius of the circle.

In this problem we have

r = 7 ft

substitute

The circumference of a circle = 2πr = 14π

2π radians subtends the complete circle of length 14π ft.

In order to find the radian measure of the central angle subtended by an arc with 4 ft length.

2π/14π = x /4

x = 8/14

x = 0.57 radians

Part 2)

The area of a circle is equal to πr^2.

where

r is the radius of the circle

A = 49 π sq. ft

The area of the complete circle of 49 π sq. ft subtends the length of the complete circle of 14 π ft.

The area of the sector formed

49 π / 14 π = x/ 4

x = 14 sq. ft

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