Answer:
[tex] \sf Perimeter = 4x + 48 [/tex]
Step-by-step explanation:
To find the perimeter of a rectangle given the area, we first need to express the area in terms of the rectangle's side lengths.
The area ([tex]A[/tex]) of a rectangle is given by the product of its length ([tex]L[/tex]) and width ([tex]W[/tex]):
[tex]\Large \boxed{\boxed{\sf A = L \cdot W }}[/tex]
In this case, the area of the rectangle is given by the quadratic expression [tex]x^2 + 24x - 81[/tex]. We need to factor this expression to identify the dimensions of the rectangle.
[tex] x^2 + 24x - 81 [/tex]
Factoring the quadratic expression:
[tex] x^2 + (27-3)x - 81 [/tex]
[tex]x^2 + 27x -3x-81 [/tex]
[tex] x(x+27)-3(x+27)[/tex]
[tex] (x + 27)(x - 3) [/tex]
Now, we can interpret these factors in the context of a rectangle's length and width. Let:
[tex] L = x + 27 [/tex]
[tex] W = x - 3 [/tex]
The perimeter ([tex]P[/tex]) of a rectangle is given by the formula:
[tex]\Large \boxed{\boxed{\sf P = 2L + 2W }}[/tex]
Substitute the expressions for [tex]L[/tex] and [tex]W[/tex] into the perimeter formula:
[tex] P = 2(x + 27) + 2(x - 3) [/tex]
Distribute and simplify:
[tex] P = 2x + 54 + 2x - 6 [/tex]
[tex] P = 4x + 48 [/tex]
So, the perimeter of the rectangle, in terms of [tex]x[/tex], is:
[tex]4x + 48[/tex]
