Expert: Ahmed Salami - 4/7/2009

Question

A university has decided to build a new, creative dorm!  After a contest to determine the shape, it was decided that the dorm would be in the shape of an airplane hangar that has semicircular ends.  There is an additional stipulation: the volume of the dorm must be exactly 225,000 cubic feet.

Part 1
The university is in the planning stages with the architects now, and they would obviously like to minimize the cost of the building.  This is where the university needs your help.  Currently, the construction costs for the foundation are $30 per square foot, the sides (the two ends) cost $20 per square foot to construct, and the roofing cost $15 per square.  The university needs your expert advice on what the dimensions of the building should be to minimize the total cost.

Determine the optimal dimensions   and  and the minimized cost, under these conditions.  The final dimensions should be rounded to two decimal places and the cost to the nearest cent.

Part 2
While the cost of flooring and siding (for the ends) has been fairly stable, a further complication factor is that the cost of roofing material has been fluctuating dramatically for quite a while.  In addition to your recommendation (in Part 1) when the price of roofing is $15 per square foot, the university also needs a recommendation on the dimensions of the dorm if the roofing costs $R per square foot.

Determine the optimal dimensions   and   under these conditions


ps. the dorm has the shape of a cylinder cut vertically in half. the distance from one semicircle to the other is y. the radious of the semicircle is x and the diameter is 2x


Answer
Hi Sayra,
Part 1
This is what i understand (correct me if i'm wrong). The volume of the building should be exactly 225000 cubic feet, the foundation area refers to the area on which the building stands, the sides are the semi-circles, the roofing is the curved surface area.
The diagram below shows the solution. Notice that i have used variables different from x and y but are defined clearly in the diagram. Adjust values for your required rounding-offs.
http://www.allexperts.com/user.cgi?m=17&imgID=13411

Part 2
See other attached diagram.

You can always get back to me.

Regards