Question

you are tasked with designing a rectangular packaging box for a model airplane. the model airplane has a length of 14 inches, a width of 8 inches, and a height of 5 inches. the box should allow for 1 inch of clearance on all four sides of the base, and 2 inches of extra height for protective padding. the material for the box costs $0.10 per square inch. determine the surface area of the smallest box that can hold the model airplane and the total cost of the material.
the surface area of the box should be 444 square inches. the total cost of the material would be $44.40.
the surface area of the box should be 606 square inches. the total cost of the material would be $60.60.
the surface area of the box should be 684 square inches. the total cost of the material would be $68.40.
the surface area of the box should be 632 square inches. the total cost of the material would be $63.20.

Explanation:

Step1: Calculate the dimensions of the box

• Length: \(14 + 1+1=16\) inches (1 - inch clearance on both sides)
• Width: \(8 + 1+1 = 10\) inches (1 - inch clearance on both sides)
• Height: \(5+2 = 7\) inches (2 - inches extra height)

Step2: Use the surface - area formula for a rectangular box \(S = 2(lw+lh+wh)\)

$$
LATEXBLOCK0
$$

Step3: Calculate the cost

Cost \(C=0.10\times S\)

$$ C = 0.10\times684=68.40 $$

Answer:

The surface area of the box should be \(684\) square inches. The total cost of the material would be \(\$68.40\). So the third option is correct.