pre calc question

Question

A farmer has 360 meters of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? What is the largest area?

Answer 1

The area is based on the length and width of the fenced area, which depends on the fencing material he has.

 

Since he only has to fence 3 sides of the field, the perimeter of the field is width + length + width, or P = 2W + L

 

We know that the perimeter will equal 360 meters, so:

360 = 2W + L

or 

360 - 2W = L

 

The area of a rectangle is width * length, or A = W*L

 

Since L = 360 - 2W we can rewrite our area equation as 

 

A = W (360-2W)

 

If we multiply this out we get

 

A = 360W - 2W²

 

If we rearrange this equation we get

 

2W² -360W + A = 0

 

Now we have a quadratic equation. The max/min of a quadratic is at the vertex. To find the x axis point of the vertex we use the equation 

 

X = -b/2a

 

Which in this case will be the width of the field.

 

Width = -(-360)/2(2) = 360/4 = 90

 

So our width at the maximum area will be 90 meters. You can use the other equations to find the length and area of the field from there.