Answer:
A_max = 11250 ft^2
Step-by-step explanation:
You have the following function for the area of the backyard:
[tex]A=x(300-2x)=300x-2x^2[/tex] (1)
To find the maximum area, you first derivative the function A respect to x:
[tex]\frac{dA}{dx}=\frac{d}{dx}[300x-2x^2]=300-4x[/tex] (2)
Next, you equal the function (2) to zero in order to obtain the value of x:
[tex]300-4x=0\\\\x=75[/tex]
Finally, you replace the value of x=75 in the function A in (1):
[tex]A=(75)(300-2(75))=11250\ ft^2[/tex]
hence, the maximum area is 11250 ft^2
