Question

Problem: Steve likes to entertain friends at parties with “wire tricks.” Suppose he takes a piece of wire 60 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. Where should Steve cut the wire so that the total area of the circle and square combined is as small as possible? What is this minimal area? What should Steve do if he wants the combined area to be as large as possible?

Need help.. Please..

Answer 1

The total length of the wire is 60 inches.
Call one piece is x inches long.
The other piece is 60 - x inches long

Useful formulas:
Area of circle = (pi)r^2
Circumference of circle = 2(pi)r
Area of square = s^2
Perimeter of square = 4s

Let's say that the piece that is x inches long is used for the circle. Since C = 2(pi)r, then r = C/(2pi), but C, the circumference is x, so
r = x/(2pi)
Acircle = (pi)r^2
Acircle = (pi)( (x)/(2pi) )^2 = ( (pi)(x^2) )/( 4(pi^2) ) = x^2/(4pi)

For the square, the perimeter is 60 - x, but since P = 4s, s = P/4, and P = 60 - x, so
s = (60 - x)/4
Asquare = s^2 = ( (60 - x)/4 )^2 = (60 - x)^2/16

The total area is Acircle + Asquare, so
A = x^2/(4pi) + (60 - x)^2/16

Answer 2

thanks

Answer 3

wlcm