A school is planning to construct two rectangular play areas in the playground.
The length of play area A must be 1 foot longer than four times its width. The width of play area B must be 2 feet longer than the width of play area A, and the length must be 2 feet longer than three times its own width. In addition, the areas of the two play areas must be equal.
Write a system of equations to represent this situation, where y is the area of the play areas and x is the width of play area A. Which statement describes the number and viability of the system’s solutions?
A.
The system has only one solution, and it is viable because it results in a positive width.
B.
The system has only one solution, but it is not viable because it results in a negative width.
C.
The system has two solutions, but only one is viable because the other results in a negative width.
D.
The system has two solutions, and both are viable because they result in positive widths.
The system of equations can be represented as follows:
Play area A:
Length = 4x + 1
Width = x
Play area B:
Length = 3(2 + x) + 2 = 6 + 3x + 2 = 8 + 3x
Width = x + 2
The areas of the two play areas must be equal, so we can set up the equation:
Area of A = Area of B
Length of A * Width of A = Length of B * Width of B
(4x + 1)(x) = (8 + 3x)(x + 2)
Expanding and simplifying:
4x^2 + x = 8x^2 + 19x + 16
Bringing all terms to one side:
0 = 4x^2 + 19x + 16 - x - 8x^2
Combine like terms:
0 = -4x^2 + 18x + 16
Divide through by -2 to simplify:
0 = 2x^2 - 9x - 8
This is a quadratic equation. To determine the solutions, the discriminant must be calculated:
Discriminant = b^2 - 4ac
= (-9)^2 - 4(2)(-8)
= 81 + 64
= 145
Since the discriminant is positive, there are two real solutions.
Now let's solve the quadratic equation to find the values of x:
2x^2 - 9x - 8 = 0
Using factoring, completing the square, or the quadratic formula, the solutions are x = -1 and x = 8/2 = 4.
However, we are interested in positive widths, so x = -1 is extraneous and not viable.
Therefore, the correct statement is:
C. The system has two solutions, but only one is viable because the other results in a negative width.