Question

3. A farmer has a 150 feet of fencing and wants to construct 3 pig pens by first building a fence around a rectangular region, then subdividing the region into 3 smaller rectangles by placing 2 fences parallel to one side of the rectangle. What dimensions of the region maximizes the total area?

Answer 1

Answer:

$l = 37.5\,ft$ and $w = 18.75\,ft$

Step-by-step explanation:

The total perimeter and area of the pig pens are, respectively:

$4\cdot w + 2\cdot l = 150\,ft$

$A = w\cdot l$

In order to know the maximum area, the second function shall be simplified, derived and equalized to zero. The solution leads to a critical point, whose characteristics can be deduced by using the second derivative of the area:

$w = 37.5\,ft - 0.5\cdot l$

$A = (37.5\,ft-0.5\cdot l)\cdot l$

$A = 37.5\cdot l - 0.5\cdot l^{2}$

$A' = 37.5 - l$

$37.5 - l = 0$

$l = 37.5\,ft$

$A'' = -1$ (absolute maximum)

$w = 37.5\,ft - 0.5\cdot (37.5\,ft)$

$w = 18.75\,ft$

The dimensions of the region that maximizes the total area are:

$l = 37.5\,ft$ and $w = 18.75\,ft$

Answer 2

Answer:

Check the explanation

Step-by-step explanation:

First note down the area equation, that is A = L * W where "A" represents the area, "L" represents the length while "W" stands for the width of a rectangle. To Solve the area equation: 30 = L * 6. Then divide the two sides of the equation by 6, and then you write down the answer. It will look like this: 5 = L.

kindly check the attached image below to get the step by step solution to the question above.