a) The perimeter function of the rectangle is [tex]p = 18\cdot x -2\cdot x^{3}[/tex].
b) The domain of the perimeter function is [tex]x \in [-3, 3][/tex].
a) The perimeter of a rectangle ([tex]p[/tex]) is the sum of the lengths of its four sides:
[tex]p = AB\cdot AC[/tex] (1)
If we know that [tex]AB = 2\cdot x[/tex] and [tex]AC = 9-x^{2}[/tex], then the perimeter of the rectangle is represented by the following formula:
[tex]p = 2\cdot x \cdot (9-x^{2})[/tex]
[tex]p = 18\cdot x -2\cdot x^{3}[/tex]
The perimeter function of the rectangle is [tex]p = 18\cdot x -2\cdot x^{3}[/tex]. [tex]\blacksquare[/tex]
b) The domain of the function is the set of values of [tex]x[/tex] associated to the function. After a quick inspection, we find that the domain of the perimeter function is [tex]x \in [-3, 3][/tex]. [tex]\blacksquare[/tex]
The statement is incomplete and poorly formatted. The correct form is described below:
As shown at the right, rectangle ABCD has vertices C and D on the x-axis and vertices A and B on the part of the parabola [tex]y = 9-x^{2}[/tex] that is above the x-axis. a) Express the perimeter [tex]p[/tex] of the rectangle as a function of the x-coordinate of A. b) What is the domain of the perimeter function?
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