Thread: Optimization (ii)

Once again this question isn't really optimization but the second part is (which I know how to do). Its the first part i'm stuck at.

A semi-circular window of rcm is divided into sectors, each of area 125cm^2

Show that the perimeter of each sector is P = 2(r + 125/r)

Thanks

Hello, r_maths!

A semi-circular window of radius r cm is divided into sectors, each of area 125 cm²

Show that the perimeter of each sector is: .P .= .2(r + 125/r)

The perimeter is: .P .= .2r + s, where s is the arclength.

Since arclength is given by: .s .= .rθ, where θ is the central angle,
. . we have: .P .= .2r + rθ .[1]


The area of a sector is given by: .A .= .½r²θ

So we have: .½r²θ .= .125 . → . θ = 250/r² .[2]

Substitute [2] into [1]: .P .= .2r + r(250/r²) .= .2r + 250/r

Therefore: .P .= .2(r + 125/r)

Thanks