Answer to Question #169917 in Calculus for Nics

Question

A farmer plans to fence his rectangular lot to secure his plantation. The lot is bounded at the back by a river; hence, no fence is needed along this side. In the front, the farmer wants to have a 24-ft opening. He surveyed the cost of the fence and noted that the fence along the front costs Php 75 per ft and Php 50 per ft along the sides. His budget for the fence is Php 15,000. 

As an architect, you were asked to prepare a plan for the fence. The plan is expected to present the 

dimensions of the largest lot that can be fenced given the budget.

Answer 1

Let $x=$ the length of the rectangular lot and $y=$ the width of the lot.

Then

$$75(x-24)+50(2y)\leq1500$$

$$0<y\leq15-0.75(x-24)$$

$$0<y\leq33-0.75x$$

Area of the rectangular lot is

$$Area=A=xy$$

$$A(x)=x(33-0.75x), 24\leq x<44$$

Find the first derivative with respect to $x$

$$A'(x)=(33x-0.75x^2)'=33-1.5x$$

Find the critical number(s)

$$A'(x)=0=>33-1.5x=0=>x=22$$

Critical number $x=22.$

Find the second derivative with respect to $x$

$$A''(x)=(33-1.5x)'=-1.5<0$$

The function $A$ has a local maximum with value of $363$ at $x=22.$

If $x>22, A'(x)<0, A(x)$ decreases.

$A(24)=24(33-0.75(24))=360(ft^2)$

The function $A$ has the absolute maximum with value of $363\ ft^2$ at $x=24\ ft.$

$y=33-0.75(24)=15(ft)$

The length is 24 ft, the width is 15 ft. The area is 360 ft².