Question

A farmer decides to enclose a rectangular​ garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 60 ft of​ fence? What should the dimensions of the garden be to give this​ area?

The maximum area that the farmer can enclose with 60ft is _____ sq feet

The larger dimension of the garden to give this area is ______ and the smaller dimension is _____

Answer 1

The perimeter of a rectangle is



Since one side is formed from the side of the barn, this means that we can take out one length (or width, it doesn't matter) to get

Plug in the given perimeter 60 (since he only has 60 ft of fencing)



The area of any rectangle is

Plug in



From now on, let's think of as where y is the area and x is the width.

Now the equation is in the form of a quadratic which has a vertex that corresponds with the maximum area. So if we find the y-coordinate of the vertex, we can find the max area.

In order to find find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)
To find the axis of symmetry, use this formula:



From the equation we can see that a=-2 and b=60

Plug in b=60 and a=-2


So the axis of symmetry is
So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.
Lets evaluate

Plug in




So the vertex is (15,450)
This shows us that the max area is then 450 square feet.
So with a width of 15 ft the fence will have a maximum area of 450 square feet
Now plug in


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Answer:

The maximum area that the farmer can enclose with 60ft is _450_ sq feet

The larger dimension of the garden to give this area is 30_ and the smaller dimension is 15