The difference of the area of two squares is 223 square feet and the difference of their perimeters is 24 feet. Find a side of each square.
Let X stand for the side of one square and Y stand for the other's side. The area is found by muliplying the two sides and the perimeter if found by adding the 4 sides of each square.
X^2-Y^2=223
4X-4Y=24
Therefore:
4X-24=4Y and dividing bothe sides by 4 gives you
X-6=Y
Substitute the X-6 for Y in the first equation and solve for X. Put that value for X in the second equation and solve for Y. Check by putting both values in the first equation.
I hope this helps. Thanks for asking.
Let's assume that the side length of the first square is "x" feet, and the side length of the second square is "y" feet.
We are given two pieces of information:
1. The difference of the area of the two squares is 223 square feet:
Area of the first square - Area of the second square = 223
x^2 - y^2 = 223
2. The difference of their perimeters is 24 feet:
Perimeter of the first square - Perimeter of the second square = 24
4x - 4y = 24
Now we have a system of equations:
x^2 - y^2 = 223 (Equation 1)
4x - 4y = 24 (Equation 2)
To solve this system, we can use the method of substitution. Rearrange Equation 2 to solve for x or y:
4x - 4y = 24
4(x - y) = 24
x - y = 6
x = 6 + y (Equation 3)
Substitute Equation 3 into Equation 1:
(6 + y)^2 - y^2 = 223
36 + 12y + y^2 - y^2 = 223
12y = 223 - 36
12y = 187
y = 187/12
y ≈ 15.58
Substitute the value of y back into Equation 3:
x = 6 + 15.58
x ≈ 21.58
Therefore, the side length of the first square is approximately 21.58 feet, and the side length of the second square is approximately 15.58 feet.
To solve this problem, let's assign variables to represent the unknowns. Let's say the side length of the larger square is x and the side length of the smaller square is y.
We are given two pieces of information:
1) The difference of the area of two squares is 223 square feet:
The area of a square is calculated by multiplying the length of one side by itself. Therefore, the area of the larger square is x * x = x^2, and the area of the smaller square is y * y = y^2. We know that the difference between these two areas is 223 square feet: x^2 - y^2 = 223.
2) The difference of their perimeters is 24 feet:
The perimeter of a square is calculated by multiplying the length of one side by 4. Therefore, the perimeter of the larger square is 4x, and the perimeter of the smaller square is 4y. We know that the difference between these two perimeters is 24 feet: 4x - 4y = 24.
Now we have a system of two equations:
Equation 1: x^2 - y^2 = 223
Equation 2: 4x - 4y = 24
To solve this system, we can use the method of substitution or elimination. Let's solve it using the elimination method:
Multiply Equation 2 by (x + y):
4x(x + y) - 4y(x + y) = 24(x + y)
4x^2 + 4xy - 4xy - 4y^2 = 24x + 24y
4x^2 - 4y^2 = 24x + 24y
Notice that this equation is equivalent to Equation 1: x^2 - y^2 = 223. So, we can rewrite equation 1 as:
4x^2 - 4y^2 = 223
Since both equations are now equal, we can cancel out the terms:
4x^2 - 4y^2 - (x^2 - y^2) = 223 - 223
3x^2 - 3y^2 = 0
3(x^2 - y^2) = 0
(x^2 - y^2) = 0
Now, we have x^2 - y^2 = 0, which means x^2 = y^2. Taking the square root of both sides, we get x = y.
Substituting x = y back into Equation 2, we have:
4x - 4x = 24
0 = 24
We have reached an inconsistency because the equation simplifies to 0 = 24, which is false. Therefore, there is no solution to this problem.
Hence, we cannot find the side lengths of the squares that satisfy the given conditions.