Find the minimum cost of a rectangular box of volume 110 cm3 whose top and bottom cost 5 cents per cm2 and whose sides cost 5 cents per cm2.

To find the minimum cost of a rectangular box with a volume of $110 \text{ cm}^3$ and given cost constraints, we need to establish the relationship between the dimensions of the box and its cost. Let the dimensions of the box be $x$, $y$, and $z$.

Given:
- Volume of the box $V = x \cdot y \cdot z = 110 \text{ cm}^3$
- Cost of the top and bottom (each of area $xy$) = 5 cents per $\text{cm}^2$
- Cost of the sides (each of areas $xz$ and $yz$) = 5 cents per $\text{cm}^2$

The total surface area $A$ of the box is given by:
\[ A = 2xy + 2xz + 2yz \]

The total cost $C$ of the box is:
\[ C = 5 \cdot (2xy + 2xz + 2yz) = 10(xy + xz + yz) \]

We need to minimize $C$ subject to the volume constraint $xyz = 110$.

Let's use the method of Lagrange multipliers. Define:
\[ f(x, y, z) = 10(xy + xz + yz) \]
\[ g(x, y, z) = xyz - 110 \]

The Lagrangian function is:
\[ \mathcal{L}(x, y, z, \lambda) = 10(xy + xz + yz) + \lambda (xyz - 110) \]

To find the critical points, we take the partial derivatives and set them to zero:

\[ \frac{\partial \mathcal{L}}{\partial x} = 10(y + z) + \lambda yz = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial y} = 10(x + z) + \lambda xz = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial z} = 10(x + y) + \lambda xy = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial \lambda} = xyz - 110 = 0 \]

From the first three equations, we get:
\[ \lambda yz = -10(y + z) \]
\[ \lambda xz = -10(x + z) \]
\[ \lambda xy = -10(x + y) \]

Let's solve these equations. Suppose $x = y = z$ for simplicity:

\[ x^3 = 110 \]
\[ x = \sqrt[3]{110} \]

Then the sides are $x = y = z = \sqrt[3]{110}$.

Now, substitute back to find the total cost. Since $x = \sqrt[3]{110}$:

\[ A = 2(\sqrt[3]{110}\sqrt[3]{110}) + 2(\sqrt[3]{110}\sqrt[3]{110}) + 2(\sqrt[3]{110}\sqrt[3]{110}) = 6(\sqrt[3]{110})^2 \]

Therefore, the total cost is:
\[ C = 10 \cdot 6(\sqrt[3]{110})^2 = 60(\sqrt[3]{110})^2 \]

\[ \sqrt[3]{110} = \approx 4.791 \]
\[ (\sqrt[3]{110})^2 \approx 22.95 \]

So the cost $C$ is approximately:
\[ C = 60 \cdot 22.95 = 1377 \text{ cents} = \$13.77 \]

Thus, the minimum cost of the rectangular box is approximately \$13.77.

Do you want details or have any questions?

Here are 5 questions you might ask next:
1. How do you solve optimization problems using Lagrange multipliers?
2. What are some common methods to find the minimum or maximum of a function?
3. How can we verify if a critical point is a minimum or a maximum?
4. Can you explain the derivation of the Lagrangian function in more detail?
5. How do volume and surface area constraints affect optimization problems?

**Tip:** When solving optimization problems, always check the constraints and ensure that the solution satisfies all of them.

Learn how to minimize the cost of a rectangular box with a volume of 110 cm^3. Explore the application of Lagrange multipliers in optimization problems involving volume and surface area constraints. Detailed step-by-step solution provided.

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