Question

Please answer this correctly

Answer 1

`\boxed{Area \: of \: quarter \: circle = 7.065 \: square \: feet}`

Given:

Perimeter of quarter circle = 10.71 feet

To find:

Area of quarter circle

Explanation:

First we need to calculate the radius of quarter circle:

Let the radius of quarter circle be 'r'

`Perimeter \: of \: quarter \: circle = (\pi r)/(2) + 2r`

`\implies 10.71 = (\pi r)/(2) + 2r \\ \\ \implies 10.71 = (\pi r)/(2) +2r (2)/(2) \\ \\ \implies 10.71 = (\pi r)/(2) + (4r)/(2) \\ \\ \implies 10.71 = (\pi r + 4r)/(2) \\ \\ \implies 10.71 * 2 = \pi r + 4r \\ \\ \implies 21.42 = \pi r + 4r \\ \\ \implies 21.42 = (\pi + 4)r \\ \\ \implies 21.42 = (3.14 + 4)r \\ \\ \implies 21.42 = 7.14r \\ \\ \implies 7.14r = 21.42 \\ \\ \implies r = (21.42)/(7.14) \\ \\ \implies r = 3 \: ft`

`Area \: of \: quarter \: circle = \frac{\pi {r}^(2) }{4} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{\pi * {(3)}^(2) }{4} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (\pi * 9)/(4) \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (3.14 * 9)/(4) \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (28.26)/(4) \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =7.065 \: {ft}^(2)`

Answer 2

So if we know the perimeter of the circle we can find it's radius using the formula for perimeter:

`p = 2\pi(r)`

So we can solve for radius:

`r = (10.71)/(2\pi)`

Then we can plug this radius into the formula for the area of a circle:

`a = \pi {r}^(2)`

`a = \pi( (10.71)/(2\pi) ) ^(2)`

Then it only wants a quarter of that area so we divide that value by 4 which upon simplification becomes the answer:

`2.28 {ft}^(2)`