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

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