Answer:
a) The polynomial is equivalent to [tex]A = (x-15)^{2}[/tex], and the side length of the playground is [tex]x -15[/tex] feet.
b) The perimeter of the playground, measured in feet, is [tex]p = 4\cdot (x-15)[/tex].
Step-by-step explanation:
a) By Geometry, we know that area of the square is equal to the square of the side length. If [tex]A = x^{2}-30\cdot x +225[/tex] is the area of the square, then must be a perfect square trinomial, that is:
[tex](x-a)^{2} = x^{2}-2\cdot a\cdot x + a^{2}[/tex] (1)
Then, we must observe the following properties:
[tex]-2\cdot a = -30[/tex] (2)
[tex]a = 15[/tex]
[tex]a^{2} = 225[/tex] (3)
[tex]a = 15[/tex]
Therefore, the polynomial is equivalent to [tex]A = (x-15)^{2}[/tex], and the side length of the playground is [tex]x -15[/tex] feet.
b) The perimeter of a square equals four times the side length. That is:
[tex]p = 4\cdot (x-15)[/tex]
The perimeter of the playground, measured in feet, is [tex]p = 4\cdot (x-15)[/tex].
The polynomial that represents the side length is x - 15, and the perimeter of the playground is 4x - 60
The expression of the area is given as:
[tex]A = x^2 - 30x + 225[/tex]
Expand the expression
[tex]A = x^2 - 15x - 15x + 225\\[/tex]
Factorize
[tex]A= (x - 15)^2[/tex]
The area of a square is:
[tex]A = l^2[/tex]
Where l represents the side length.
So, we have:
[tex]l = x - 15[/tex]
The perimeter is then calculated as:
[tex]P = 4l = 4*( x - 15)[/tex]
[tex]P = 4x - 60[/tex]
Hence, the perimeter of the playground is 4x - 60
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