Pls help me!!!!! C2 maths-sectors/segments

2) A minor sector OMN of a circle has centre O and radius r cm. The perimeter of the sector is 100cm and the area of the sector is A cm^2.
a) Show that A=50r-r^2.
b) Given that r varies, find:
i) The value of r for which A is a maximum and show that A is a maximum.
ii) The value of angle MON for this maximum area.
iii) The maximum area of the sector OMN.

5) A sector OAB of a circle, centre O and radius r cm. The length of the arc AB is p cm and angle AOB is theta radians.
a) Find theta in terms of p and r.
b) Deduce that the area of the sector is ½ pr cm^2.

Reply 1

Fermat

The sector has two sides of length = r and an arc of length s = rθ
So, perimeter of sector is given by,

p = 2r + rθ
========

Area of a sector is gven by the formula,

A = ½r²θ
=======

a)
We are told the the perimeter, p = 100

p = 100 = 2r + rθ
rθ = 100 - 2r

substituting for rθ in the formula for the area of a sector,

A = ½r.rθ
A = ½r.(100 - 2r)
A = 50r - 2r²
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b i)
A = 50r - 2r²
dA/dr = 50 - 2r, = 0 for a turning point
50 - 2r = 0
r = 25 cm
=======

d²A/dr² = -2
Therefore, at r = -2 (which is where the turning point occurs), A is a maximum since d²A/dr² is -ve.

b ii)
p = 100 = 2r + rθ

at r = 25,

100 = 50 + 25θ
θ = 2 radians
==========

b iii)
A = ½r²θ
A = ½.(25)².2
A = 625 cm²
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