# Maximum Area

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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 -

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 -

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 -

As the text drawing did not come out clear I hope this may clarify what I meant.

imgur /F5NJVqp

• Maximum Area -

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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