The area A of a rectangular garden is given by the expression2^2+ 2 − 12. The length L of the garden is
given by the expression x2 - 9. Find an expression for the width W of the garden. (Recall that A = LW). What
are the excluded values in this calculation and what do they represent in context?

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kudzordzifrancis kudzordzifrancis · Answer 1 · 2020-04-15T04:24:25+02:00

Answer:

[tex]W=\frac{2(x - 2)}{ (x - 3)}[/tex]

Step-by-step explanation:

We have that the area of a rectangular garden is

[tex]A = 2 {x}^{2} + 2x - 12[/tex]

and the length of the rectangular garden is

[tex]L = {x}^{2} - 9[/tex]

We want to find an expression for the width W of the garden.

From A=LW, we have

[tex]W=\frac{A}{L}[/tex]

[tex]W=\frac{2 {x}^{2} + 2x - 12}{ {x}^{2} - 9}[/tex]

[tex]W=\frac{2 ({x}^{2} + x - 6)}{ {x}^{2} - 9}[/tex]

We factor to get:

[tex]W=\frac{2 (x +3)(x - 2)}{ (x - 3)(x + 3)}[/tex]

Since this is a rational function, there is a hole at x=-3 and a vertical asymptote at x=3. These are the excluded values.

[tex]W=\frac{2(x - 2)}{ (x - 3)}[/tex]