Question

the length, ( l ), of a rectangle is modeled by the equation ( l = w + 4 ), where ( w ) is the width of the rectangle in centimeters. two equations have been determined that represent the area of the rectangle, ( a ), in square centimeters:

• the first equation was created using the formula for the area of a rectangle: ( a = w^{2}+4w ).
• the second equation models the relationship between the rectangles area and width: ( a = 4w + 45 ).

which statement describes the solution(s) of the system?
a. there are two solutions, and both are viable.
b. there are two solutions, but only one is viable.
c. there is only one solution, and it is viable.
d. there are two solutions, and neither are viable.

Explanation:

Step1: Set the two equations equal

Since \(A = w^{2}+4w\) and \(A = 4w + 45\), we set \(w^{2}+4w=4w + 45\).
Subtract \(4w\) from both sides:
\(w^{2}+4w-4w=4w + 45-4w\)
\(w^{2}=45\)

Step2: Solve for \(w\)

Take the square root of both sides: \(w=\pm\sqrt{45}=\pm3\sqrt{5}\approx\pm6.71\)
But width \(w>0\) (because width cannot be negative in the context of a rectangle's dimension). So \(w = 3\sqrt{5}\approx6.71\) is viable and \(w=- 3\sqrt{5}\approx - 6.71\) is not.

Answer:

B. There are two solutions, but only one is viable.