A trash company is designing an open-top, rectangular container that will have a volume of $1080 \mathrm{ft}^{3}$. The cost of making the bottom of the container is $\$ 5$ per square foot, and the cost of the sides is $\$ 4$ per square foot. Find the dimensions of the container that will minimize total cost. \[ \mathrm{L} \times \mathrm{W} \times \mathrm{H}=\square \mathrm{ft} \times \square \mathrm{ft} \times \square \mathrm{ft} \]
Solution
Step 1 ：Let the length, width, and height of the container be \(L\), \(W\), and \(H\) respectively. The volume of the container is given by \(L \times W \times H = 1080 \, \mathrm{ft}^{3}\).
Step 2 ：Since the container is open-top, the total cost of making the container is the sum of the cost of the bottom and the cost of the sides. The bottom is a rectangle with area \(L \times W\), so its cost is \(5LW\). Each side is a rectangle with area \(LH\), \(WH\), or \(LH\), so the total cost of the sides is \(4LH + 4WH\). Therefore, the total cost \(C\) is given by \(C = 5LW + 4LH + 4WH\).
Step 3 ：We want to minimize \(C\) subject to the constraint \(LWH = 1080\). To do this, we can use the method of Lagrange multipliers. We form the Lagrangian function \(L = 5LW + 4LH + 4WH - \lambda(LWH - 1080)\), where \(\lambda\) is the Lagrange multiplier.
Step 4 ：We take the partial derivatives of \(L\) with respect to \(L\), \(W\), \(H\), and \(\lambda\), and set them equal to zero. This gives us the system of equations \(\frac{\partial L}{\partial L} = 0\), \(\frac{\partial L}{\partial W} = 0\), \(\frac{\partial L}{\partial H} = 0\), and \(\frac{\partial L}{\partial \lambda} = 0\).
Step 5 ：Solving this system of equations, we find that the dimensions that minimize the total cost are \(L = \sqrt[3]{\frac{1080}{2}}\) ft, \(W = \sqrt[3]{\frac{1080}{2}}\) ft, and \(H = 2\sqrt[3]{\frac{1080}{2}}\) ft. Therefore, the dimensions of the container that will minimize total cost are \(\boxed{L = \sqrt[3]{\frac{1080}{2}} \, \mathrm{ft}, W = \sqrt[3]{\frac{1080}{2}} \, \mathrm{ft}, H = 2\sqrt[3]{\frac{1080}{2}} \, \mathrm{ft}}\).
