Final answer:
The cubic equation to find the length of Fabiana's rectangular metal box, given it must have a volume of 150 in.³, would be x³ - 3x² + 2x - 150 = 0, where x is the length of the box.
Step-by-step explanation:
Fabiana is constructing a rectangular metal box with a given volume of 150 in.³, a length of 2 inches, and where the width is 3 inches less than the length, and the height is 1 inch more than the length. To find the length of the box, we can set up a cubic equation based on the volume formula for a rectangular prism, which is length × width × height. Since the width is 2 inches - 3 inches and the height is 2 inches + 1 inch, we substitute these expressions into the volume formula:
Volume = length × (length - 3) × (length + 1).
Plugging in the volume Fabiana wants, we get:
150 = 2 × (2 - 3) × (2 + 1).
However, since we know the length is 2 inches and we are trying to find an equation that would help Fabiana find the correct length, we should define the variable x as the length. The cubic equation would then be:
x³ - 3x² + 2x - 150 = 0.
This cubic equation represents the relationship between the length of the box and the volume it must have.