Answer:
Part a)
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
we have
substitute
solve for W
<u><em>Find the area of the rectangular playground</em></u>
The area is given by
we have
substitute
Convert to function notation
Part b) What side length x gives the maximum area that the playground can have?
we have
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
Factor -1
Complete the square
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
The playground is a square
