Math problem help...minimum cost.

[emptywallet]

I've been sitting here for awhile and I can't figure this out. For some reason its extremely confusing to me, I'm assuming there are two different functions that you need to right, and then possibly find the derivative of both, but I just can't get it right. The problem is under and application of Maxima and Minima : Minimum cost, if that helps any. Here's the problem:

***"From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $5 dollars per foot and the fence for the middle costs $2 dollars per foot. If each lot contains 13,500 square feet, find the dimensions of each lot that yield the minimum cost for the fence. ***"


I hate word problems, but if the problem is stated for me I can get it pretty easily, its just forming the problem that is hard, especially from something like this. Anyone help?

 

[Mdguy]

I beat Fonz to this?

OK, let's see:

Draw 2 areas A and B side by side sharing a middle boundary . Since the Areas are equal in size and share a side, their lengths (x) are equal to one another (it's hard w/o seeing the diagram)

Area A=Area B = 13,500= xy

Cost:

Cost of Fence in $ = $2/ft (2y +4x) + $5/ft
Cost = 9y + 8x

you want to minimize COST, so you have to convert it into an equation of one variable, differentiate it, and set the result equal to zero:

Cost = 9(13500)/x + 8x

dCost/dx = -9(13500)/x squared + 8 = 0

x squared = 9(13500)/8

solve for x....find y...and then substitute into your inital cost equation. That's it. I HOPE I didn't make any math mistakes along the way, but I'm sure someone will correct me if I did!

 

[Mdguy]

WHOOPS! I just glanced at this again and saw that I made an error-I transposed the costs of the fencing. The equation should be Cost of Fence in $ = $5/ft (4x + 2y) + $2/ft (2y)
= 20x +12y
get to one variable, take the derivative, etc. and you should have it. Sorry. Where was Fonz or Warik to correct me?

 

[Fonz]

WHOOPS! I just glanced at this again and saw that I made an error-I transposed the costs of the fencing. The equation should be Cost of Fence in $ = $5/ft (4x + 2y) + $2/ft (2y)
= 20x +12y
get to one variable, take the derivative, etc. and you should have it. Sorry. Where was Fonz or Warik to correct me?

Yes, you copied it down wrong.

The fence is rectangular.

So, its(2y+2x)*5(outside fence) + 2(middle fence)

Therefore, its 12y + 20x

Area(xy) = 13,500

Therefore, y = 13,500/x

Cost = 12(13,500/x) + 20x

Cost = 162,000/x + 20x

Take the derrivativeAnd set it equal to zero)

0 = - 162 000/x^2 + 20

162 000 = 20x^2

x^2 = 8100

x = 90

Therefore if X = 90 then y = 13,500/90 = 150

The plot is therefore 150 by 90.(A rectangle)

Fonz