Question

a piece of pizza in the shape of a sector with angle, theta, and radius, r, is to be cut that has perimeter equal to 20 inches. how should r and theta be chosen so that the area of the piece is as large as possible. A=(1/2)(r^2)(theta)

Answer 1

Given that perimeter of piece of pizza is 20 inches, So

Perimeter = P = 2r + s

P = 2r + r* = 20 inch

= (20 - 2r)/r = 20/r - 2

Now area of piece of pizza will be:

A = (1/2)*r^2*

A =(1/2)*r^2*(20/r - 2)

A = 10*r - r^2

Now Area will be maximum when, dA/dr = 0

dA/dr = d[10r - r^2]/dr

dA/dr = 10*1 - 2*r = 0

10 = 2r

r = 5

And

= 20/r - 2 = 20/5 - 2 = 2 rad

So for maximum possible area

radius of piece = 5 inch, = 2 rad

Let me know if you've any query.