To solve question 3, we need to minimize the cost of constructing a 12-cubic-foot box with five wooden sides and a glass front. The cost of wood is $5 per square foot, and the cost of glass is $10 per square foot.

Given that:

• The volume of the box is 12 cubic feet.
• The glass and the wooden side opposite the glass are both $2 \times 2$ feet.
• The other four sides are $2 \times 3$ feet.

Step 1: Volume Constraint

We know the box has a total volume of 12 cubic feet. Since the glass and its opposite side are both $2 \times 2$, the remaining dimensions can be set as follows:

$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$

From the dimensions given:

• Two sides are $2 \times 2$ feet, which implies one side (the height) is 3 feet for the $2 \times 3$ side. Thus, the volume calculation aligns with the volume of the box: $2 \times 2 \times 3 = 12 \, \text{cubic feet}$

Step 2: Surface Area Calculation

• Glass side: $2 \times 2 = 4$ square feet
• Opposite wooden side: $2 \times 2 = 4$ square feet
• Four remaining wooden sides: $2 \times 3 = 6$ square feet each

Thus, the total surface area is: $\text{Wood surface area} = 4 \, (\text{opposite}) + 4 \times 6 = 4 + 24 = 28 \, \text{square feet of wood}$ $\text{Glass surface area} = 4 \, \text{square feet}$

Step 3: Cost Calculation

Now, calculate the total cost based on the area and the cost per square foot:

• Wood cost: $28 \, \text{sq. ft.} \times 5 \, \text{\$/sq. ft.} = 140 \, \text{dollars}$
• Glass cost: $4 \, \text{sq. ft.} \times 10 \, \text{\$/sq. ft.} = 40 \, \text{dollars}$

Thus, the total cost is: $140 + 40 = 180 \, \text{dollars}$

Conclusion:

The dimensions that minimize the cost are:

• Glass side: $2 \times 2$ feet
• Opposite wooden side: $2 \times 2$ feet
• Other wooden sides: $2 \times 3$ feet

The total cost of materials is $180.

Do you need any further details or clarifications on this solution?

Here are 5 related questions to expand on this:

1. What would happen to the cost if both the glass and wood had the same price per square foot?
2. How would the dimensions change if the total volume was 24 cubic feet instead of 12?
3. Can we minimize the cost even further if the dimensions are flexible, but the volume is fixed?
4. How would adding an extra glass panel to one of the $2 \times 3$ sides affect the cost?
5. What if the cost of wood increased by 20%? How would this impact the optimal dimensions?

Tip: Always check if optimizing dimensions for cost still satisfies any volume constraints before finalizing your answer.