Answer:
[tex]625 m^2[/tex]
Step-by-step explanation:
In this problem, we are told that the area of the garden is given by the expression
[tex]A(w)=-(w-25)^2+625[/tex]
where
w is the width of the garden (in meters)
Here we want to find the maximum possible area.
The maximum of a function f(x) can be found by requiring that its first derivative is zero:
[tex]f'(x)=0[/tex]
Therefore, here we have to calculate the derivative of [tex]A(w)=0[/tex] and find the value of w for which it is equal to zero.
Let's start by rewriting the area function as
[tex]A(w)=-(w^2-50w+625)+625=-w^2+50w[/tex]
Now we calculate the derivative with respect to w:
[tex]A'(w)=-2w+50[/tex]
Now we require this derivative to be zero, so
[tex]-2w+50=0\\w=-\frac{50}{-2}=25 m[/tex]
So now we can substitute this value of w into the expression of A(w) to find the maximum possible area:
[tex]A(25)=-(25-25)^2+625 = 625 m^2[/tex]
This value is allowed because we know that the maximum length of the perimeter of the fence is 100 meters; If the garden has a square shape, the length of each side is [tex]L=\frac{100}{4}=25 m[/tex], and the area of the squared garden is
[tex]A=L^2=(25)^2=625 m^2[/tex]
Which is equal to what we found earlier: this means that the maximum area is achieved if the garden has a squared shape.
