SOLUTION: 4) 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

Question 131108: 4) 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?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Remember 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)

Subtract from both sides

Rearrange the equation

Now let's introduce another formula. The area of any rectangle is

Plug in

Rearrange the terms

Distribute

Rearrange the terms

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

Multiply 2 and -2 to get -4

Reduce

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

Start with the given polynomial

Plug in

Raise 15 to the second power to get 225

Multiply 2 by 225 to get 450

Multiply 60 by 15 to get 900

Now combine like terms

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

Multiply

Subtract

-------------------------------

Answer:

So the dimensions of the garden are

width: 15, length: 30

Also, the max area of the garden is 450 square feet.
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