Question

A farmer has 300 ft of fencing with which to enclose a rectangular pen next to a barn. The barn itself will be used as one of the sides of the enclosed area.



What is the maximum area that can be enclosed by the fencing?

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ft²

Answer 1

Let
y------> the length of the rectangle
x-----> the width of the rectangle

we know that
perimeter of the rectangle=2*[x+y]

Since the barn is used as one of the sides (let's say y) we can subtract y
we don't need fencing for this side
That makes the perimeter 2x + y
Since we have 300 feet of fencing
we set these equal:
 2x + y = 300-------> y=(300-2x)------> equation 1

area of the rectangle=x*y
substitute the equation 1 in the area formula
Area=[(300-2x)]*x-----> Area=(-2x²)+300x

This is a quadratic equation
Since the leading coefficient is negative (-2) we know it opens downward 
We are looking for the x-coordinate of the highest point called the vertex

using a graph tool
see the attached figure

the vertex is the point (75,11250)
that means for x=75 (width of the rectangle)

the area is 11250 ft²

the answer is
the maximum area that can be enclosed by the fencing is 11250 ft²

Answer 2

Answer:

The Answer is 11250 ft~

Step-by-step explanation:

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