Tuesday, September 2, 2014

Suppose you want to fence off two rectangular pieces of land that have the same area. You have 300 feet of fencing. The total length of both pieces of land are 2x feet (each x feet in length). The width of the land is y feet (2 y's on the outside representing the width of the overall and and y feet in width down the middle separating the two pieces of land)

What is the area of the land in terms of x?  What length and width will give the maximum area of the land?

You know that area of a rectangular is length times width.  So the overall area is x(y). But we want the entire area in terms of x.

Well we know the total amount of fencing is 300 feet, so x + x + y + y + y = 300. (just took all the dimensions and added them).

2x + 3y = 300

Solve for y to get 3y = 300 - 2x

and y = 100 - (2/3)x

So now we can get the area all in terms of x..

Area = x(100 - (2/3)x)

100x - (2/3)x^2

To find the value of x which maximizes area, you can use calculus or simply take -b/2a, where b is the number in front of the x and a is the number in front of the x^2.

So -b/2a = -100/[2(-2/3)] = 75

The length is 75. To find the length, substitute 75 in for x in the equation 2x + 3y = 300. Therefore, y = 50.

The width of the land is 50 feet for a total area of 3,750 square feet.

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