Final Answer:
If the perimeter of a square is a inches and the area of the square is b square inches, and the sum of a and b is equal to 45 then the length of the sides of the square is 5 inches
Step-by-step explanation:
We know that the perimeter (P) of the square is a inches and the area (A) of the square is b square inches.
We are also told that the sum of a and b is equal to 45, which can be written as:
a + b = 45 (Equation 1)
Understand the properties of a square:
A square has 4 equal sides. Let's denote the length of one side of the square as s inches.
The perimeter (P) of a square is the sum of all its sides, which can be calculated by:
P = 4s
The area (A) of a square is the side length squared, which can be calculated by:
A = s^2
Set up the equations based on what we know:
a = 4s (Equation 2)
b = s^2 (Equation 3)
Combine the equations with the given sum of perimeter and area:
Substitute Equation 2 and Equation 3 into Equation 1:
4s + s^2 = 45
This is a quadratic equation that we can solve for s.
Rearrange the quadratic equation:
s^2 + 4s - 45 = 0
Factor the quadratic equation:
To solve for s, we need to find factors of -45 that add up to 4.
The factors of -45 are:
(-1, 45), (-3, 15), (-5, 9), (1, -45), (3, -15), (5, -9)
From these pairs, (5, -9) are the factors we need because 5 + (-9) = -4.
However, since we are looking for a sum of +4 in the quadratic equation, we use the reverse signs:
(s + 9)(s - 5) = 0
Solve for s:
s + 9 = 0 or s - 5 = 0
From this point, we solve for s from each equation:
s = -9 or s = 5
Since a negative side length doesn't make sense for a square, we disregard s = -9.
Therefore, the length of one side of the square (s) is 5 inches.