A piece of string 40 cm long is cut into two lengths. Each length is laid out to form a square.

Question

A piece of string 40 cm long is cut into two lengths. Each length is laid out to form a square. Given that the length of the sides of one of the squares is xcm, find expressions in terms of x for (a) the length of the sides of the other square (b) the total area enclosed by the two squares. Given also that the sum of the two areas is a minimum, (c) calculate the value of x.

Answer 1

40 cm string is cut in 2 pieces. each piece forms a square

if one square has sides = x each, what is the side of the other square??

perimeter of that 1st square = 4x, string left for 2nd square = 40-4x, side of 2nd square = (40-4x)/4 = 10-x

a) side of 2nd square = 10-x cm

b) total area of the 2 squares = x^2 + (10-x)^2 = 100-20x+2x^2 cm^2

c) minimum area is when both squares are the same size and x=5, Min area = 50 cm^2

area of the 2 squares is a parabola formula with min = vertex

in vertex form 2x^2 -20x +100 = 2(x^2 -10 +25)+ 100- 50= (2(x-5)^2 +50, with vertex = (5,50)

side of 5 cm makes minimum area of 50 cm^2