Another algebra question! A farmer decides to enclose a rectangular garden, ?

Question

using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 80 ft. of fence? What should the dimensions of the garden be to give this area? Any and all responses are appreciated!

Answer 1

Ok because the fence is beside the barn, there will be only 3 sides of fencing. This means that there are 2 widths and one length (because the other side of the length is the barn wall) So:

Let the width = y

Let the length = x

So you know that the farmer has 80ft of fence (i.e. the perimeter of the fence) So:

2y + x = 80

Also, you know that the area of a rectangle is length x width:

A = lw or in this case xy

If you rearrage the perimeter equation, you get:

2y + x = 80

x = 80 - 2y

Now you know that x = 80 - 2y. You know SUBSTITUTE x in the area equation for this equation:

A = xy

A = (80-2y)y

A = 80y - 2y^2   (Multiply it out)

A = -2y^2 + 80y

Now that you have that, I'm not sure if you know this but you have to complete the square. First you do is factor out the 2 and:

A = -2(y^2 - 40y)

A = -2(y^2 - 40y + 200 - 200) (divide the 40 by 2 and square it) You need the plus 200 and minus 200 to keep the equation balanced (because they cancel eachother out)

A = -2(y - 20)^2 + 800 (trinomial factoring and multiply the -200 our of the brackets)

So know you know that the maximum area you can get is 800 based on the last number in the final equation. You also know that y = 20 (your goal in the brackets is for all of it to equal zero and y has to equal 20 in order for the brackets to equal zero.

So now that you know that y = 20, you can figure x from the first equation way above.

2y + x = 80

2(20) + x = 80

40 + x = 80

x = 80-40

x = 40

Therefore, the dimensions are 40 x 20ft with a maximum area of 800ft

Answer 2

depends on how long the barn is.

Say the barn was 40 ft.



80 ft of fence plus 40 ft of barn equals 120 ft perimeter.

since the barn is 40 ft, the opposite  side has to be 40 ft.to make a rectangle

80 ft of fence minus 40 ft on one side div by 2 equals each short side is 20 ft.

Therefore the garden is 20 by 40 or 800 sq ft

If the barn were 20 ft, then the sides would be 30 ft and the garden 600 sq ft

if the barn were 10 ft, the sides are 35 ft and the garden 350 sq ft.

Get it?

Answer 3

I am just giving my best shot here...with the information given...this is the best I can come up with:

Area = LW

If you have 80ft of total fence, you'd need to divide 80 by 3...that will give you the maximum distance of each side:

          80/3 = 26.67ft (the longest distance the 3 sides can be)

Since Area = LW, multiply 26.67 X 26.67 to get the total area:

         Maximum Area = 26.67 X 26.67 = 711.2889ft^2