A 100 inch piece of wire is cut into two pieces. Each piece of wire is used to make a square wire frame. Let x be the length of one piece of the wire. Determine an algebraic representation A(x) for the total area of the two squares.
The two pieces are of length x and 100-x
Since a square has four equal sides,
A(x) = (x/4)^2 + ((100-x)/4)^2
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To determine an algebraic representation A(x) for the total area of the two squares, we need to express the length of one side of each square in terms of x.
Let's denote the length of one side of the first square as a, and the length of one side of the second square as b.
Given that x represents the length of one piece of the wire, the total length of wire used for both squares is 100 inches.
Hence, we can construct the following equation to represent the total length of wire used:
2a + 2b = x
Since each square has four equal sides, the perimeter of each square is 4a and 4b respectively. Therefore, the equation can be simplified as follows:
4a + 4b = x
Dividing both sides of the equation by 4 gives us:
a + b = x/4
Now, to find the area of a square, we square the length of one side. Hence, the area of the first square is a^2, and the area of the second square is b^2.
To find the total area, we need to add the areas of both squares:
A(x) = a^2 + b^2
We now have the equations:
a + b = x/4 (1)
A(x) = a^2 + b^2 (2)
To express A(x) solely in terms of x, we need to find a way to eliminate a and b from equation (2).
To do this, we'll square equation (1):
(a + b)^2 = (x/4)^2
Simplifying,
a^2 + 2ab + b^2 = x^2/16
Now, we can substitute this expression for a^2 + b^2 in equation (2):
A(x) = x^2/16 - 2ab
Hence, the algebraic representation for the total area of the two squares, A(x), is:
A(x) = x^2/16 - 2ab
To determine an algebraic representation A(x) for the total area of the two squares, we need to find the side length of each square using the given information.
Let's assume that the length of one piece of the wire is x. Since the wire is cut into two pieces, the length of the other piece will be (100 - x) because the sum of the lengths of the two pieces should be equal to the length of the original wire (100 inches).
Now, we need to find the side length of each square. The perimeter of a square is equal to 4 times the side length. In this case, the perimeter of each square is equal to the length of each corresponding piece of wire.
So, for the square made from the wire of length x, the perimeter is x, and for the square made from the wire of length (100 - x), the perimeter is (100 - x).
We know that the perimeter of a square is 4 times the side length. So, for the square with perimeter x, the side length can be calculated as (x / 4). Similarly, for the square with perimeter (100 - x), the side length will be ((100 - x) / 4).
Now that we have the side lengths of both squares, we can calculate their areas.
The area of a square is given by the formula A = side length * side length. So, the area of the first square (with side length x/4) will be (x/4)^2, and the area of the second square (with side length (100 - x)/4) will be ((100 - x)/4)^2.
Finally, we can add the areas of both squares to get the total area A(x) of the two squares:
A(x) = (x/4)^2 + ((100 - x)/4)^2
That is the algebraic representation for the total area of the two squares.