a wire of 10 cm is cut into pieces forming two squares of different sizes. what is the maximum and minium area of the squares?
The wire that is 10 cm long is cut into two pieces each of which is used to form a square. If the length of one piece is x cm, the other piece is (10 - x) cm long.
The sum of the area of the two pieces is` A = (x/4)^2 + ((10 - x)/4)^2`
= `x^2/16 + 100/16 + x^2/16 - 20x/16`
=> `x^2/8 + 25/4 - 5x/4`
The derivative of A is `A' = x/4 - 5/4`
`x/4 - 5/4 = 0`
=> x = 5
The minimum area of the squares is 3.125 cm^2 and the maximum area is 6.25 cm^2.
The minimum area will be when the 10 cm wire is cut in to 2 equal pieces of 5 cm. In this case length of the side of the square will be equal to (10/2)/4 =5/4cm and the combined area will be 2*(5/4)^2 = 25/8 = 3.125cm2
The maximum area of the two squares will be when one square is made of zero length and the other from 10cm. In this case the side of the larger square will be equal to 10/4 = 5/2cm and area will be (5/2)^2 = 6.25cm2
The maximum areas of the squares are 6.25cm2 and
the minimum areas of the squares are 3.125cm2