Question

what is the area of sector of a circle with radius 4cm when perimeter of sector is 20cm?​

Answer 1

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• Let OAB be the given sector of a circle of radius r cm such that AB=1 cm and ∠AOB=θ radius then

Area of sector OAB=20 cm

$⇒2r+l=20⟶(1)$

• Let S be the area of sector OAB. Then

$S= 21​ r 2 θS= 21​ r 2 × rl​ = 21​ lrS= 201​ (20−2r)rS=10r−r 2$

This is the required expression for the area S if the sector in term of r

$S=10r−r 2 drdS​ =10−2rdr 2 d 2 S​ =−2$

• For maximum / minimum value of S we must have

$dS​ =0⇒10−2r=0r=5cm Clearly dr 2 d 2 S​ < for all r. S is maximum where r=10Maximum value of S= 10×5−(5) 2 =25cm 2 .$