Plz solve this problem..
A wire of 100 cm is cut into two parts and each part is bent to form square.if the sum of the areas of two squares is 425 cm square.find the length of side of two square.
if the squares have sides x and y, then
4x+4y = 100 or, x+y=25
x^2+y^2 = 425
My vast experience tells me instantly that 425 = 400 + 25 = 20^2+5^2
Let's assume the length of one part of the wire is x cm. Therefore, the length of the other part will be 100 - x cm.
Now, let's calculate the perimeter of each square, which is equal to the sum of the lengths of all sides.
For the first square:
Perimeter = 4 * (length of one side) = 4 * x
For the second square:
Perimeter = 4 * (length of one side) = 4 * (100 - x)
We know that the sum of the areas of the two squares is 425 cm².
So, the equation becomes:
x² + (100 - x)² = 425
Expanding the equation:
x² + (100 - x)(100 - x) = 425
x² + (10000 - 200x + x²) = 425
2x² - 200x + 10000 - 425 = 0
2x² - 200x + 9575 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 2, b = -200, and c = 9575.
Calculating the discriminant:
Discriminant = b² - 4ac
= (-200)² - 4 * 2 * 9575
= 40000 - 76600
= -36600
Since the discriminant is negative, the equation has no real solutions. This means that it is not possible to cut the wire into two parts and form two squares with a total area of 425 cm².
To solve this problem, we can break it down into smaller steps:
Step 1: Assign variables
Let's assign a variable to represent the length of one part of the wire that is bent to form a square. We'll call this variable "x". Since the wire is cut into two parts, the length of the other part would be (100 - x).
Step 2: Formulate equations
Now we can use this information to write equations. The perimeter of a square is equal to four times the length of its side. Since the wire is bent to form two squares, the equations become:
4x + 4(100 - x) = 100 (Equation 1) -> 4x + 400 - 4x = 100
Area of the first square: x^2
Area of the second square: (100 - x)^2
The sum of the areas of both squares is given as 425. So we have the equation:
x^2 + (100 - x)^2 = 425 (Equation 2)
Step 3: Solve the equations
Now we can solve the equations simultaneously to find the value of "x" (the length of one side of the square).
First, simplify Equation 1:
4x + 400 - 4x = 100
400 = 100
We see that this equation leads to a contradiction (400 ≠ 100), which means there is no solution to this system of equations. Therefore, there is no length of side that satisfies the given conditions.
Hence, there is no solution to this problem.