A 10 meter long rope is to be cut into two pieces and a square is to be made using each. The difference in the areas enclosed must be 1 1/4 square meters how should it be cut?
To solve this problem, we need to find the lengths of the two pieces of the rope that will create squares with areas that differ by 1 1/4 square meters.
Let's assume that one piece of the rope has a length of "x" meters. This means the other piece will have a length of "10 - x" meters.
The area of a square is equal to the side length squared. So, the area of the first square will be x^2 square meters, and the area of the second square will be (10 - x)^2 square meters.
Now, we can set up our equation:
(x^2) - ((10 - x)^2) = 1 1/4
To simplify the equation, let's expand (10 - x)^2:
(x^2) - (100 - 20x + x^2) = 1 1/4
Cancel out common terms:
-100 + 20x = 1 1/4
Rearrange the equation:
20x = 101 1/4
Now, let's convert 1 1/4 to an improper fraction:
20x = 125/4
Multiply both sides by 4 to get rid of the fraction:
80x = 125
Divide both sides by 80 to solve for x:
x = 125/80 = 1.5625
So, one piece of the rope should be approximately 1.5625 meters long. Since the total length is 10 meters, the other piece would be approximately 10 - 1.5625 = 8.4375 meters long.
Therefore, the rope should be cut into two pieces, one approximately 1.5625 meters long and the other approximately 8.4375 meters long, in order to create squares with areas that differ by 1 1/4 square meters.
To solve this problem, let's assume that one piece of the rope is "x" meters long. The other piece would then be "10 - x" meters long.
Now, let's consider the two squares that would be made from these pieces. The area of a square is given by multiplying its side length by itself.
The first square would have a side length of "x/4" meters, since the perimeter of the square is equal to the length of the rope piece. Therefore, the area of the first square would be (x/4)^2 = x^2/16 square meters.
The second square would have a side length of "(10 - x)/4" meters, since the perimeter of the square is equal to the length of the second rope piece. Therefore, the area of the second square would be ((10 - x)/4)^2 = (10 - x)^2/16 square meters.
According to the problem, the difference in the areas of these two squares must be 1 1/4 square meters. Therefore, we can set up the following equation:
x^2/16 - (10 - x)^2/16 = 1 1/4
Simplifying this equation, we get:
x^2/16 - (100 - 20x + x^2)/16 = 5/4
x^2 - (100 - 20x + x^2) = 20
-20x + 100 = 20
-20x = -80
x = 4
So, one piece of the rope should be 4 meters long, and the other piece should be 10 - 4 = 6 meters long.
If the sides are x and y, then
4x+4y = 10
x^2 - y^2 = 5/4
Now just find x and y