Let’s solve the problem step by step.

Step 1: Define Variables

Let:

• $w$ be the width of the base of the box (in cm).
• $l$ be the length of the base of the box (in cm). Given that the length is twice the width, we have $l = 2w$.
• $h$ be the height of the box (in cm).

Step 2: Volume Equation

The volume $V$ of the box is given by:

$V = l \cdot w \cdot h = 2w \cdot w \cdot h = 2w^2h$

We know the volume is 3500 cc, so:

$2w^2h = 3500$

Solving for $h$:

$h = \frac{3500}{2w^2} = \frac{1750}{w^2}$

Step 3: Surface Area Equation

The surface area $A$ of the open-top box (no lid) is given by:

$A = lw + 2lh + 2wh$

Substituting $l = 2w$:

$A = (2w \cdot w) + 2(2w \cdot h) + 2(w \cdot h) = 2w^2 + 4wh + 2wh = 2w^2 + 6wh$

Substituting $h = \frac{1750}{w^2}$:

$A = 2w^2 + 6w \left(\frac{1750}{w^2}\right) = 2w^2 + \frac{10500}{w}$

Step 4: Minimize the Surface Area

To find the height that minimizes the surface area, we need to find the critical points of $A$. We do this by taking the derivative of $A$ with respect to $w$ and setting it to zero:

$\frac{dA}{dw} = 4w - \frac{10500}{w^2}$

Set the derivative equal to zero to find the critical points:

$4w - \frac{10500}{w^2} = 0$

$4w^3 = 10500$

$w^3 = \frac{10500}{4} = 2625$

$w = \sqrt[3]{2625}$

Calculating $w$:

$w \approx 13.64 \, \text{cm}$

Step 5: Find the Height $h$

Now, use the value of $w$ to find $h$:

$h = \frac{1750}{w^2} = \frac{1750}{(13.64)^2}$

Calculating $h$:

$h \approx 9.4 \, \text{cm}$

So, the height $h$ that makes the surface area as small as possible is approximately $9.4$ cm.

Would you like more details or have any questions?

Related Questions:

1. How do you find the critical points of a function?
2. How does the width of the base affect the surface area of the box?
3. What would happen if the box had a lid? How would that change the calculations?
4. How do you use the second derivative test to confirm if a critical point is a minimum or maximum?
5. How does changing the volume of the box affect the optimal height?

Tip: For optimization problems, always ensure your function is correctly defined and includes all constraints before finding the critical points!