Answer-
The maximum area that they can fence off is 6400 ft²
Solution-
Organizers of an outdoor concert will use 320 feet of fencing to fence off a rectangular vip section.
i.e the perimeter of the rectangular section is 320 feet
Let us assume,
x = length of the rectangular section
y = breadth of the rectangular section
Hence,
[tex]\Rightarrow 2(x+y)=320\\\\\Rightarrow x+y=160\\\\\Rightarrow y=160-x[/tex]
Now, we have to find the maximum area for which they can fence that off.
The area of the rectangular section is,
[tex]=x\cdot y[/tex]
So we have to maximize the area function.
[tex]\Rightarrow f(x)=x\cdot y[/tex]
Putting the value of y,
[tex]\Rightarrow f(x)=x\cdot (160-x)=160x-x^2[/tex]
[tex]\Rightarrow f'(x)=160-2x[/tex]
[tex]\Rightarrow f''(x)=-2[/tex]
Finding the critical values,
[tex]\Rightarrow f'(x)=0[/tex]
[tex]\Rightarrow 160-2x=0[/tex]
[tex]\Rightarrow 2x=160[/tex]
[tex]\Rightarrow x=80[/tex]
∵ f"(x) is negative (i.e -2), so for the value of x=80, f(x) or area function will be maximum.
[tex]f(x)_{\text{at x=80}}=f(80)=160(80)-(80)^2=12800-6400=6400[/tex]
Therefore, the maximum area that they can fence off is 6400 ft²
