Question

if a sector of a circle has area 100².Find the perimeter of the sector

Answer 1

Answer:

Let t° be the angle of sector at the centre of the circle.

Area of the sector = π.r^2.(t/360°). cm^2 = 100.cm^2.

or, (t/360°) = 100/(π.r^2)……………..(1).

Perimeter (p) of the sector = 2.r + 2.π.r.(t/360°). , putting (t/360°)=100/π.r^2.

or, p = 2.r +2.π.r.(100)/π.r^2.

or, p = 2.r + 200.(1/r). ………………(2)

or, dp/dr = 2 +200.(-1/r^2). , for maximum or minimum putting dp/dr =0.

or, 0. = 2 - 200/r^2.

or, r^2 = 100. => r = +/- 10. , r = 10 . r = -10 (not valid).

d2p/dr^2 = 400/r^3. , d2p/dr^2 is +ve , there exist minimum at x=10.

Minimum value of the perimeter = 2.r+200.(1/r) = 2×10 + 200×(1/10). = 40.cm.

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