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

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Remember the perimeter of a rectangle is

P=2W%2B2L

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


P=2W%2BL


60=2W%2BL Plug in the given perimeter 60 (since he only has 60 ft of fencing)

60-2W=L Subtract 2W from both sides


L=60-2W Rearrange the equation



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

A=L%2AW


A=%2860-2W%29%2AW Plug in L=60-2W


A=W%2A%2860-2W%29 Rearrange the terms


A=60W-2W%5E2 Distribute


A=-2W%5E2%2B60W Rearrange the terms


From now on, let's think of A=-2W%5E2%2B60W as y=-2x%5E2%2B60x 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:

x=-b%2F%282a%29

From the equation y=-2x%5E2%2B60x we can see that a=-2 and b=60

x=%28-60%29%2F%282%2A-2%29 Plug in b=60 and a=-2


x=%28-60%29%2F-4 Multiply 2 and -2 to get -4



x=15 Reduce


So the axis of symmetry is x=15


So the x-coordinate of the vertex is x=15. Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate f%2815%29

f%28x%29=-2x%5E2%2B60x Start with the given polynomial


f%2815%29=-2%2815%29%5E2%2B60%2815%29 Plug in x=15


f%2815%29=-2%28225%29%2B60%2815%29 Raise 15 to the second power to get 225


f%2815%29=-450%2B60%2815%29 Multiply 2 by 225 to get 450


f%2815%29=-450%2B900 Multiply 60 by 15 to get 900


f%2815%29=450 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



L=60-2%2815%29 Now plug in w=15


L=60-30 Multiply


L=30 Subtract


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Answer:

So the dimensions of the garden are

width: 15, length: 30

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