A farmer has 160 feet of fencing to enclose 2 adjacent rectangular pig pens. What dimensions should be used so that the enclosed area will be a maximum?

Answer 1

I'm assuming that the pig pens have identical dimensions.

Let's assume that the pig pens need to be fenced in the way shown in the diagram above.

Then, the perimeter is given by #4x + 3y = 160#.

#4x = 160 - 3y#

#x = 40 - 3/4y#

The area of a rectangle is given by #A = L xx W#, however here we have two rectangles put together, so the total area will be given by #A = 2 xx L xx W#.

#A = 2(40 - 3/4y)y#

#A = 80y - 3/2y^2#

Now, let's differentiate this function, with respect to y, to find any critical points on the graph.

#A'(y) = 80 - 3y#

Setting to 0:

#0 = 80 - 3y#

#-80 = -3y#

#80/3 = y#

#x = 40 - 3/4 xx 80/3#

#x = 40 - 20#

#x = 20#

Hence, the dimensions that will give the maximum area are #20# by #26 2/3# feet.

A graphical check of the initial function shows that the vertex is at #(26 2/3, 1066 2/3)#, which represents one of the dimensions that will give the maximum area and the maximum area, respectively.

Hopefully this helps!

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Answer 2

To maximize the enclosed area, the pens should be squares with sides of 40 feet each.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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