Question

You have a wire that is 89 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?

Answer 1

assume the square side = x, then its circumference = 4x

so the circle circumference would be 89 - 4x which equals 2pi r

so x = 1/4 (89 - 2pi r)

now A(r) = x^2 + pi r^2

= [1/4 (89 - 2pi r)]^2 + pi r^2

diff. w.r.to r u get:

dA/dr = 1/8 (89-2 pi r)(-2 pi) + 2 pi r = 0

2pi r - pi/4 (89-2 pi r) = 0

2r - (89 - 2 pi r)/4 = 0

2r + pi r /2 = 89/4

mult. the eqn by 4 and solve for r u get:

r = 89 / (8 + 2pi) cm

and the circle circumference = 2 pi r = 2 pi 89 / (8 + 2pi)

= 178 pi / (8 + 2pi) = 89 pi /(4 + pi) cm

hope this helps