A piece of wire 44 cm long is cut into two parts and each part is bent to form a square. If the total area of the two squares is 65 sq cm, how do you find the perimeter of the two squares?
1 Answer
Here's what I got.
Explanation:
You know that you're working with a piece of wire that is
44 - x -> the second piece
Now, these pieces are used to form two squares. Since a square has four equal sides, the length of one side of the first square will be
Similarly, the length of one side of the second square will be
The area of a square is given by
color(blue)(|bar(ul(color(white)(a/a)"area" = "side"^2color(white)(a/a)|)))
In your case, the area of the first square will be
A_1 = (x/4)^2 = x^2/16
The are of the second square is
A_2 = ((44-x)/4)^2 = (44 -x )^2/16
The problem tells you that the total area of the square
A_"total" = A_1 + A_2
is equal to
65 = x^2/16 + (44-x)^2/16
This is equivalent to
x^2 + (44-x)^2 = 65 * 16
x^2 + 44^2 - 88x + x^2 = 1040
Rearrange to quadratic equation form
2x^2 - 88x +896 = 0
This quadratic has two solutions as given by the quadratic formula
x_(1,2) = (-(-88) +- sqrt( (-88)^2 - 4 * 2 * 896))/(2 * 2)
x_(1,2) = (88 +- sqrt(576))/4
x_(1,2) = (88 +- 24)/4 implies {(x_1 = (88 + 24)/4 = 28), (x_2 = (88 - 24)/4 = 16) :}
Here comes the cool part. You know that the sides of the two squares are
"For the 1"^("st")" square: "28/4" "color(red)("or")" "16/4" "
"For the 2"^("nd")" square: "(44-28)/4 = 16/4" " color(red)("or")" "(44-16)/4 = 28/4 " "
As you know the perimeter of a square is given by the equation
color(blue)(|bar(ul(color(white)(a/a)"perimeter" = 4 xx "side" color(white)(a/a)|)))
This means that the perimeters of the two squares are
"For the 1"^("st") " square: "4 xx 28/4 = "28 cm " color(red)("or") " "4 xx 16/4 = "16 cm"
"For the 2"^("nd")" square: " 4 xx 16/4 = "16 cm " color(red)("or") " "4 xx 28/4 = "28 cm"
This means that if the perimeter of the first square is
Similarly, if if the perimeter of the first square is
