Question

-.) Farmer S. Unkist has a fruit grovec fruit do not mix before they are pro . Unkist has a fruit arave consisting of lemons, bananas, and watermelons along a moat. To prevent thieves from stealing his fruit while at the same time make sure the o not mix before they are processed, he plans to fence in each fruit plot using identical ctangular enclosures. The side along the moat needs no fence because the moat is with man eating crocodiles as shown below. If he has 1200 yards of fence, what ve the dimensions of each enclosure if the total area of the grove is to be maximized? WORK USE CALCULUS TECHNIQUES TO OBTAIN YOUR ANSWER AND SHOW ALL

Answer 1

Answer

The farmer plans to fence each of the three fruit plots with identical rectangle enclosures.

Let the dimensions of the each rectangle be x by y, where x side of the rectangle is parallel to the moat.

N

The farmer has 1200 yards of fence. Thus, the constraint is

$3x+4y = 1200\Rightarrow x=\frac{1200-4y}{3}$

The objective is to maximize the total area covered by the rectangles.

The total area covered by the three rectangles is

$A(y)=3xy=3\left ( \frac{1200-4y}{3} \right )y=(1200-4y)y=1200y-4y^2$

To maximize the function A(y), first we have to find the critical point(s) by setting A'(y) to 0 and then we will do the second derivative test.

$A'(y)=\frac{\mathrm{d} }{\mathrm{d} y}(1200y-4y^2) =1200(1)-4(2y)=1200-8y$

$A'(y)=1200-8y=0\Rightarrow y=\frac{1200}{8}=150$

Now,

$A''(y)=\frac{\mathrm{d} }{\mathrm{d} y}(1200-8y)=0-8(1)=-8$

As A''(150) = -8 < 0, A(y) is maximum at y = 150.

The x value is

$x=\frac{1200-4y}{3}=\frac{1200-4(150)}{3}=200$

Hence, the dimensions of each enclosure should be 200 yards by 150 yards to maximize the total area of the grove.