To solve the problem of finding the sides of the two squares given their area and perimeter, we can use algebraic expressions.
Given Information
- The sum of the areas of the two squares: 57 m²
- The sum of their perimeters: 68 m
Let the sides of the squares be a and b.
The area of a square is given by the formula:
- Area of first square: \( a^2 \)
- Area of second square: \( b^2 \)
Thus, the equation for the areas becomes:
- \( a^2 + b^2 = 57 \) (Equation 1)
The perimeter of a square is given by the formula:
- Perimeter of first square: \( 4a \)
- Perimeter of second square: \( 4b \)
Thus, the equation for the perimeters becomes:
- \( 4a + 4b = 68 \)
- Simplifying gives: \( a + b = 17 \) (Equation 2)
Solving the Equations
Now we have a system of equations:
1. \( a^2 + b^2 = 57 \)
2. \( a + b = 17 \)
From Equation 2, we can express \( b \) in terms of \( a \):
- \( b = 17 - a \)
Substituting this into Equation 1:
- \( a^2 + (17 - a)^2 = 57 \)
- Expanding gives:
- \( a^2 + (289 - 34a + a^2) = 57 \)
- Combining like terms results in:
- \( 2a^2 - 34a + 232 = 0 \)
- Dividing by 2:
- \( a^2 - 17a + 116 = 0 \)
Finding Roots
Using the quadratic formula:
- \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Here, \( b = -17 \), \( a = 1 \), \( c = 116 \):
- Discriminant: \( (-17)^2 - 4 \times 1 \times 116 = 289 - 464 = -175 \)
Since the discriminant is negative, there are no real solutions, indicating an error in calculations or assumptions. Please verify the initial conditions or check for an alternative approach.