Boundary Wire

Please help me solve this question in Maths? I need steps and explanations as possible?

A peice of wire, of fixed length L cm, is bent to form the boundary OPQO of a sector of a circle. The circle has centre O and radius r cm. The angle of the sector is theita radians.

a) Show that the area A cm^2 of the sector is given by:
A = 1/2 rL – r^2

b) Find a relationship between r and L for which A is greatest and find the corresponding value of theita.

c) Show, that for this value of theita, the area of triangle OPQ is approximately 45.5% of the area of the sector OPQ.

Thanks.

a) Use the pic (answer is there)

b). A = ½ rL – r²

dA/dr = ½L – 2r
dA/dr = ½ (L – 4r)
dA/dr = 0 when r = L/4

when r < L/4
dA/dr >0

when r > L/4
dA/dr < 0

So the maximum occurs when r = L/4

c) r = L/4

S = rƟ
Ɵ = (L-2r)/r
Ɵ rad = (L/2) / (L/4)
Ɵ rad = 2

pi rad = 180
2 rad = 360/pi
Ɵ = 114.6degrees

Are of POQ
A' = (1/2) (2 * r * sin(Ɵ/2)) * (rcosƟ/2)
A' = (r^2/2)sinƟ
A = (r^2/2) (0.91)

A of the sector = (1/2)rL - r^2
A = (1/2) r(4r) - r^2
A = r^2

A' / A x100% = (0.91)/2 x 100% = 45.5%

PS: drop a note if u can't get how I got the areas.

 

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