To minimize the surface area of a rectangular box with an open top, where the length is 3 times the width, we can follow these steps:

1. Define Variables:

   • Let the width be $w$.
   • Then the length $l$ is $3w$.
   • Let the height be $h$.

2. Surface Area Formula: The surface area $S$ of a box with an open top is given by: $S = l \cdot w + 2h \cdot w + 3h \cdot w$ Substituting for $l$: $S = 3w^2 + 2hw$

3. Express Height in Terms of Width: To minimize the surface area, we can express height in terms of volume if needed. However, without a specific volume, we will differentiate with respect to $w$.

4. Optimization: To find the minimum surface area, we can take the derivative of the surface area $S$ with respect to $w$ and set it to zero: $\frac{dS}{dw} = 6w + 2h$ Setting $\frac{dS}{dw} = 0$: $6w + 2h = 0 \implies h = -3w$ Since height cannot be negative, we look for additional constraints.

5. Volume Constraint: If a specific volume $V$ were given, we would express $h$ in terms of $w$ and $V$: $V = lwh = 3w^2h$ Solving for $h$: $h = \frac{V}{3w^2}$

6. Substituting Back: Substitute $h$ back into the surface area equation and minimize: $S(w) = 3w^2 + 2\left(\frac{V}{3w^2}\right)w = 3w^2 + \frac{2V}{3w}$

7. Take Derivative Again: Find $\frac{dS}{dw}$ and set it to zero to find critical points.

This process will lead to a function that can be minimized. The specifics depend on the volume $V$.

If you can provide a specific volume, I can give you a more concrete solution with numerical values.

Do you have a volume constraint, or would you like to proceed differently?

Related Questions:

1. What is the desired volume of the box?
2. Would you like to see the numerical solution for a specific volume?
3. How does the surface area change if the length is doubled instead?
4. What would happen to the surface area if the box were closed?
5. Are there other shapes that could yield a smaller surface area for the same volume?

Tip: To minimize surface areas efficiently, always consider the relationship between the dimensions and any volume constraints you have.