Posted by **Lisa** on Sunday, March 10, 2013 at 8:44pm.

We have 900 ft of fencing and we want to construct a back yard. If we are using the building as part of the barrier for the yard what are the dimensions that allow maximum area?

I am not even sure how to start though I know I am supposed to calculate the derivative of the setup. Thank you

- Maximum Area -
**Steve**, Sunday, March 10, 2013 at 8:53pm
if the yard has dimensions x and y, and side x is parallel to the building, then the fencing used is

x+2y = 900

The area is

a = xy = (900-2y)*y = 900y - 2y^2

da/dy = 900 - 4y

da/dy=0 when y=900/4 = 225

so, x = 900-2y = 450

the yard is thus 450 by 225

- Maximum Area -
**Lisa**, Sunday, March 10, 2013 at 9:06pm
I do not think I made it clear sir. The part where I was stuck was where 100 ft. of the building is covered by the side of the yard.

___________________

| yard |

| |

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

} building {

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

the building length is 100 feet

Sorry for any confusion this caused.

- Maximum Area -
**Lisa**, Sunday, March 10, 2013 at 9:08pm
As the text drawing did not come out clear I hope this may clarify what I meant.

imgur /F5NJVqp

- Maximum Area -
**Steve**, Monday, March 11, 2013 at 1:21am
You're right - you did not make it clear that the building was only 100 feet wide.

If the building on;y covers 100 feet of the side x, then

x+(x-100)+2y = 900

2x+2y=1000

x = 500-y

and follow the same steps above.

If I still have it wrong, then just draw yourself a diagram and label the various parts, making sure all the fence sections add up to 900. Then substitute for x or y in the formula a=xy.

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