Question

You have been tasked to build a rectangular box. The top and bottom are to be built out of 2 cm thick aluminum. The density of aluminum is 2.7 g per cm^3, and the cost $6 per kg. The front and back walls are to be build out of 1 cm thick stainless steel. The density of stainless steel is 7.5 g per cm^3, and the cost $4 per kg. Finally, the 2 remaining walls are to be built out of 1 cm thick copper. The density of copper is 9.0 g per cm^3, and the cost $9 per kg. Find the measurements and total price of the least expensive such box that has volume 1 m^3

Answer 1

price:

for top and bottom:

$4xy\cdot2.7\cdot0.006=0.0648xy$

for front and back:

$2xz\cdot7.5\cdot0.004=0.06xz$

for 2 remaining walls:

$2yz\cdot9\cdot0.009=0.162yz$

Volume:

$V=xyz=10^6$ cm²

$0.0648xy+0.06xz+0.162yz\to min$

surface area:

$A=2xy+2yz+2xz$

$z=V/(xy)$

$A=2xy+2V/x+2V/y$

$\frac{\partial A}{\partial x}=2y-2V/x^2=0\implies y=2V/x^2$

$\frac{\partial A}{\partial y}=2x-2V/y^2=0\implies x=2V/y^2$

$y=2Vy^4/(4V^2)=y^4/(2V)$

$y=\sqrt[3]{2V}=\sqrt[3]{2\cdot10^6}=100\sqrt[3]{2}$ cm

$x=2\cdot10^6/(10^4\cdot\sqrt[3]{4})=100\sqrt[3]{2}$ cm

$z=10^6/(10^4\sqrt[3]{4})=100/\sqrt[3]{4}$ cm

Total price:

$P=0.0648xy+0.06xz+0.162yz=$

$=0.0648\cdot10^4\sqrt[3]{4}+0.06\cdot 10^4/\sqrt[3]{2}+0.162\cdot 10^4/\sqrt[3]{2}=\$2790.65$