Expert: Scotto - 11/18/2008

Question
Hello Scott, hopefully you can help me out with this question!

The volume of the dorm must be excatly 225,00 cubit 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 (floor) are $30 per square foot, the sides (ends) cost $20 per square foot to construct and the roofing (curved part) costs $15 per square foot.  Find the dimensions for the building that minimize the total cost of the building.

You can assume the ends are half circles as well.  

Ok, well I started out by saying x+y+z = Total Cost, where x y and z are the cost of the floor, roofing, and sides.  I got about halfway through this and took the derivitave then have no idea where to go from there.  Can you please help?

Zach

Answer
If L is thelength and W is the width, I can say the area A of the floor is A=LW and it's cost is 30LW.

Both sides of the building together make up a circle, so the area is πW²/4 and the cost is 30πW²/4 = (7.5)πW².

The roof woulld be half the side of a cylinder and would cost $15/ft².  It has a lenght L and a width that is half of the length of the perimeter of one of the circles.  The perimeter of a circle is 2πr, and here r=W/2, which means the the width is πW.  The cost on the roof is 15πW.

Our total cost is then the TC = floor+sides+wall
= 30LW + 7.5πW² + 15πW.

The total volume is half of a cylinder, so it is πr²L/2.  We also know that r is W/2.  This gives an equation for volume
22,500 = πW²L/8, which says that L=8*22,500/(πW²).

Putting this into the equation for total cost would then gives us the total cost as a function of w.

Take the derivative, set it equal to 0, and find W.

With this value of W, we can find L.  Oncw W and L are known, we know what the buidling looks like.