A wire is 264 ft in length is to be cut into two pieces, one to form a rectangle whose length is four less than twice the width, and the other to form a square whose side is equal to the sum of the width and the length of the rectangle. how long is the shorter piece of wire
Width = X Ft.
Length = (2x-4) Ft.
Square:
Each side = x + (2x-4).
2x+2(2x-4) + 4(x + 2x-4) = 264 Ft.
2x+4x-8 + 4x+8x-16 = 264.
18x - 24 = 264.
18x = 288.
X = 16 Ft.
Rectangle = 2*16 + 2(2*16-4) = 88 Ft.
Square = 4(16+32-4) = 176 Ft.
So the shorter piece of wire = 88 Ft.
To find the length of the shorter piece of wire, we need to solve the problem step by step:
1. Let's start by assigning variables to the unknown quantities. Let's say "x" represents the width of the rectangle.
2. According to the problem, the length of the rectangle is four less than twice the width. So, the length of the rectangle would be "2x - 4".
3. The wire is cut into two pieces, with one piece forming a rectangle and the other piece forming a square. The length of the square is equal to the sum of the width and the length of the rectangle. Therefore, the side length of the square would be "(x + (2x - 4))".
4. The perimeter of a square is four times its side length. So, the perimeter of the square would be "4 * (x + (2x - 4))".
5. The perimeter of the rectangle is given by the formula: "2 * (length + width)". In this case, the length is "2x - 4" and the width is "x". So, the perimeter of the rectangle would be "2 * ((2x - 4) + x)".
6. According to the problem statement, the total wire length is 264 ft. Therefore, we can set up the equation: "4 * (x + (2x - 4)) + 2 * ((2x - 4) + x) = 264".
7. Simplify the equation and solve for "x":
4x + 8x - 16 + 4x - 8 + 2x - 4 = 264
18x - 28 = 264
18x = 292
x = 16.22 (rounded to two decimal places)
8. Since we are looking for the length of the shorter piece of wire, which corresponds to the perimeter of the rectangle, substitute the value of "x" into the perimeter equation: "2 * ((2 * 16.22 - 4) + 16.22)".
9. Calculate the perimeter of the rectangle, which will give us the length of the shorter piece of wire.
By following these steps, you can find the length of the shorter piece of wire.