Let's solve this optimization problem step-by-step.

Given Information

1. The top and bottom margins of the poster are each 6 cm, so there’s a total of 12 cm added to the height.
2. The side margins of the poster are each 4 cm, so there’s a total of 8 cm added to the width.
3. The area of the printed material (the central part of the poster) is 384 cm².

We need to find the dimensions of the entire poster that result in the smallest possible area.

Step-by-Step Solution

1. Define Variables:

   • Let $w$ be the width of the printed area (in cm).
   • Let $h$ be the height of the printed area (in cm).

2. Printed Area Equation: Since the area of the printed material is given as 384 cm²: $w \times h = 384$ Therefore, $h = \frac{384}{w}$.

3. Total Dimensions of the Poster:

   • The total width of the poster would be $W = w + 8$ (adding 4 cm on each side).
   • The total height of the poster would be $H = h + 12$ (adding 6 cm on the top and bottom).

4. Poster Area Function: The total area $A$ of the poster is given by: $A = W \times H = (w + 8) \times \left(\frac{384}{w} + 12\right)$

5. Simplify the Area Function: Expand and simplify the expression: $A = w \times \frac{384}{w} + 12w + 8 \times \frac{384}{w} + 8 \times 12$ $A = 384 + 12w + \frac{3072}{w} + 96$ $A = 480 + 12w + \frac{3072}{w}$

6. Optimization: To find the minimum area, take the derivative of $A$ with respect to $w$ and set it to zero: $\frac{dA}{dw} = 12 - \frac{3072}{w^2}$ Set $\frac{dA}{dw} = 0$: $12 = \frac{3072}{w^2}$ $w^2 = \frac{3072}{12} = 256$ $w = 16$

7. Calculate $h$: Substitute $w = 16$ back into the printed area equation to find $h$: $h = \frac{384}{16} = 24$

8. Find Total Dimensions:

   • Total width $W = w + 8 = 16 + 8 = 24$ cm.
   • Total height $H = h + 12 = 24 + 12 = 36$ cm.

Conclusion

The dimensions of the poster with the smallest area are:

• Width: 24 cm
• Height: 36 cm

Would you like more details on any part of this solution?

Here are 5 related questions you might explore:

1. How does the addition of margins affect the optimization of area?
2. What other methods can be used to verify the minimum area dimensions?
3. How would the solution change if only one margin dimension was given?
4. What if the printed area was not fixed but flexible within a range?
5. How does changing the aspect ratio of the printed area affect the poster dimensions?

Tip: In optimization problems, identifying variables and relationships is essential. Set up equations carefully before applying calculus for best results.