Question

The radius of a circle is 3 kilometers. What is the area of a sector bounded by a 120° arc?Give the exact answer in simplest form. ____ square miles.

Answer 1

The exact answer in simplest form for the area of the sector is 2.258π square miles.

To find the area of a sector, you can use the formula:

Area of Sector=
`(central angle )/(360) * \pi r^2`

where:

Central Angle is the angle in degrees of the sector,

π is a mathematical constant approximately equal to 3.14159,

r is the radius of the circle.

In this case, the radius (r) is given as 3 kilometers, and the central angle is 120°.

Area of Sector=
`(120)/(360) * \pi *(3km)^2`

Now, calculate the area:

Area of Sector=
`(1)/(3) * \pi * 9km^2`

Area of Sector=3πkm^2

Now, you mentioned the answer should be in square miles. To convert square kilometers to square miles, you can use the conversion factor: 1 square kilometer = 0.239 square miles.

Area of Sector in square miles=3π×0.239mi^2

Now, calculate the numerical value:

Area of Sector in square miles≈2.258πmi^2

So, the exact answer in simplest form for the area of the sector is 2.258π square miles.

Answer 2

Ok, we know that 120° is 1/3 of 360° that is the angle total of the circunference, so the of a sector bounded by a 120° will be 1/3 of the total area.

So, we are going to use the next equation:

`\text{Area}=(1)/(3)\pi r^2`

So, replacing the radius we get:

`=(1)/(3)(3)^2\pi`
`=(1)/(3)(9)\pi`
`=(9)/(3)\pi`
`=3\pi`

Now, we are going to convert square kilometers to square miles:

`=3\pi km^2\cdot(1mile^2)/(2.59km^2)`
`\cong3.64miles^2`

The area of a sector bounded by a 120° arc is aproximately 3.64 square miles.