A heated storage room is shaped like a rectangular box and has a volume of 1000

Question

A heated storage room is shaped like a rectangular box and has a volume of 1000 cubic fee, as shown in the figure. Because warm air rises, the heat loss per unit of area through the ceiling is five times as great as the heat loss through the floor. The heat loss through the four walls is three times as great as the heat loss through the floor. Determine the room dimensions that will minimize heat loss and therefore minimize
heating costs

Explanation / Answer

Let x,y,z be the dimensions of the box, with x and y being the floor dimensions. We want to minimize H = xy + 5 * xy + 3 * (2xy + 2xz) = 6xy + 6xz + 6yz, subject to V = xyz = 1000 Using Lagrange Multipliers, ?H = ??V ==> = ? So, we have 6y + 6z = ?yz 6x + 6z = ? xz 6x + 6y = ? xy Hence, ? xyz/6 = xy + xz = xy + yz = xz + yz. ==> x = y = z. So, V = x^3 = 1000 ==> x = 10. So, the box should have the dimensions of a cube with edge length 10 ft. I hope this helps! Work done on yahoo answers by kb

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