A small, open-top packing box, similar to a shoebox without a lid, is three times as long as it is wide, and half as high as it is long. Each square inch of the bottom of the box costs $0.80 to produce, while each square inch of any side costs $0.03 to produce. If x represents the number of inches in the width of the box,which of the following functions represents the cost, C, of producing the box?

Options

A. `C(x)=0.42x^(2)`
B. `C(x)=0.60x^(2)`
C. `C(x)=0.72x^(2)`
D. `C(x)=0.96x^(2)`

Answer

To solve the problem, we need to derive a cost function \( C \) for the packing box based on its dimensions and the costs associated with producing its surfaces. Let's break it down step by step.

### Step 1: Define the dimensions of the box
Let:
- \( x \) = width of the box (in inches)
- Length \( l = 3x \) (since the length is three times the width)
- Height \( h = \frac{l}{2} = \frac{3x}{2} \) (since the height is half the length)

### Step 2: Calculate the area of the bottom of the box
The area of the bottom of the box is given by:
\[ \text{Area of the bottom} = \text{Length} \times \text{Width} = l \times x = (3x) \times x = 3x^2 \text{ square inches} \]

### Step 3: Calculate the cost of the bottom
The cost to produce the bottom is:
\[ \text{Cost of the bottom} = \text{Area of the bottom} \times \text{Cost per square inch} = 3x^2 \times 0.80 = 2.4x^2 \text{ dollars} \]

### Step 4: Calculate the area of the sides of the box
The box has four sides:
1. Two sides with dimensions \( h \times x \)
2. Two sides with dimensions \( h \times l \)

Calculating the area of the two sides with dimensions \( h \times x \):
\[ \text{Area of two sides} = 2 \times (h \times x) = 2 \times \left(\frac{3x}{2} \times x\right) = 3x^2 \text{ square inches} \]

Calculating the area of the two sides with dimensions \( h \times l \):
\[ \text{Area of two sides} = 2 \times (h \times l) = 2 \times \left(\frac{3x}{2} \times 3x\right) = 9x^2 \text{ square inches} \]

### Step 5: Total area of the sides
The total area of the sides is:
\[ \text{Total area of sides} = 3x^2 + 9x^2 = 12x^2 \text{ square inches} \]

### Step 6: Calculate the cost of the sides
The cost to produce the sides is:
\[ \text{Cost of the sides} = \text{Total area of sides} \times \text{Cost per square inch} = 12x^2 \times 0.03 = 0.36x^2 \text{ dollars} \]

### Step 7: Total cost function
The total cost \( C \) of producing the box is the sum of the costs of the bottom and the sides:
\[ C = \text{Cost of the bottom} + \text{Cost of the sides} = 2.4x^2 + 0.36x^2 = 2.76x^2 \text{ dollars} \]

### Final Answer
Thus, the function that represents the cost \( C \) of producing the box is:
\[ C(x) = 2.76x^2 \]