Answer:
Part a) [tex]A(x)=(-x^2+240x)\ m^2[/tex]
Part b) The side length x that give the maximum area is 120 meters
Part c) The maximum area is 14,400 square meters
Step-by-step explanation:
The picture of the question in the attached figure
Part a) Find a function that gives the area A(x) of the playground (in square meters) in terms of x
we know that
The perimeter of the rectangular playground is given by
[tex]P=2(L+W)[/tex]
we have
[tex]P=480\ m\\L=x\ m[/tex]
substitute
[tex]480=2(x+W)[/tex]
solve for W
[tex]240=x+W\\W=(240-x)\ m[/tex]
Find the area of the rectangular playground
The area is given by
[tex]A=LW[/tex]
we have
[tex]L=x\ m\\W=(240-x)\ m[/tex]
substitute
[tex]A=x(240-x)\\A=-x^2+240x[/tex]
Convert to function notation
[tex]A(x)=(-x^2+240x)\ m^2[/tex]
Part b) What side length x gives the maximum area that the playground can have?
we have
[tex]A(x)=-x^2+240x[/tex]
This function represent a vertical parabola open downward (the leading coefficient is negative)
The vertex represent a maximum
The x-coordinate of the vertex represent the length that give the maximum area that the playground can have
Convert the quadratic equation into vertex form
[tex]A(x)=-x^2+240x[/tex]
Factor -1
[tex]A(x)=-(x^2-240x)[/tex]
Complete the square
[tex]A(x)=-(x^2-240x+120^2)+120^2[/tex]
[tex]A(x)=-(x^2-240x+14,400)+14,400[/tex]
[tex]A(x)=-(x-120)^2+14,400[/tex]
The vertex is the point (120,14,400)
therefore
The side length x that give the maximum area is 120 meters
Part c) What is the maximum area that the playground can have?
we know that
The y-coordinate of the vertex represent the maximum area
The vertex is the point (120,14,400) -----> see part b)
therefore
The maximum area is 14,400 square meters
Verify
[tex]x=120\ m[/tex]
[tex]W=(240-120)=120\ m[/tex]
The playground is a square
[tex]A=120^2=14,400\ m^2[/tex]
