The arc length of a sector is equal to four times the radius. Express the arc length s

as a function of the area a

of the sector.

Step-by-step explanation:

Given that,

The arc length is four times the radius

Let he radius be 'r'

Then, the arc length be 's'

The arc of a sector can be calculated using

s=θ/360 * 2πr

Then, given that s=4r

So, 4r = θ * 2πr / 360

Divide both side r

4 = θ * 2π/360

Then, make θ subject of formula

θ * 2π = 360 * 4

θ = 360 * 4 / 2π

θ = 720 / π

So, area of the sector can be determine using

A = θ / 360 * πr²

Since r = ¼s

Then,

A = (θ/360) * π * (¼s) ²

A = (θ/360) * π * (s²/16)

A = θ * π * s² / 360 * 16

Since θ = 720 / π

A = (720/π) * π * s² / 360 * 16

A = 720 * π * s² / 360 * 16 * π

A = s² / 8

Then,

s² = 8A

Then,

s = √ (8A)

s = 2 √2•A