Answer:

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

Explanation :

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