Expert: Scotto - 11/23/2008

Question
the volume of the dorm must be exactly 225,000 cubic feet, which is one
cubic foot for each sprout on the Chia plane. We’re in the planning stages
with the architects now, and we would obviously like to minimize the cost of
the building. Currently, the construction costs for the foundation are $30 per
square foot, the sides cost $20 per square foot to construct, and the roofing
costs $15 per square foot. Find the dimensions for the building that
minimize the total cost of the building.

While the cost of the flooring and siding has been fairly stable, a further
complicating factor is that the cost of the roofing has been fluctuating
dramatically. In addition to your recommendation for the price of $15 per
square foot, I also need your recommendation on the dimensions of the
dorm if the roofing costs $R per square foot.
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Answer
Foundation: #30/ft²; sides: $20/ft²; and roofing: $15/ft².

The dimensions of the building are L, W, and H for length, width,
and height, respectively.

Area of the base is LW and the cost is 30,ft², so cost is 30LW.
Area of th sides are 2LH + 2WH, so cost at $20/ft² is 40(LH+WH).
Roofing is LW, so cost at $15/ft² is 15LW.

Adding all of the costs together gives [1] 45LW + 40LH + 40WH.
Since there is no variance between L and W in the equation, we will take L=W.  Putting this back in [1] gives [2] 45W² + 80WH.

We know that the volumne is suppose to be 225,000 ft².  
We also know that the volume is WLH.
Now we took L=W, so the volume equation is 225,000 = W²H.

Using this equation, we can find H -> H=225,000/W².
This can be put back into [1] to get 45W² + C/W where the constant is 80*225,000.  Since 225,000 is 1/4 of 900,000, 8*225,000=1,800,000.

What needs to be done is then to take the derivative of
45W² + 1,800,000/W, set the derivative equal to 0, and solve for W.
Note that the derivative of 1/W is -1/W², which gives a minus sign.  This term can then be added to both sides, making one positive equal the the other.  Divide by the constant (90) times W and multiply both sides by W, then take the cube root.