Final answer:
To find the dimensions of the box when the width is 3 feet, we can use the volume formula and solve for x. The equation v(x) = 2x^3 - 15x^2 + 31x - 12 represents the volume, and by substituting the width into the equation, we can find the values of x that correspond to the length and height of the box.
Step-by-step explanation:
To find the dimensions of the box, we need to determine the length and height. Since the width is given as 3 feet, we can use the formula for the volume of a rectangular box, which is V = lwh, where V is the volume, l is the length, w is the width, and h is the height.
Given that the volume is modeled by v(x) = 2x^3 - 15x^2 + 31x - 12, we need to solve for x when w = 3.
By substituting w = 3 into the equation, we have v(x) = 2x^3 - 15x^2 + 31x - 12 = 0. Using either factoring, synthetic division, or a graphing calculator, we can find the values of x, which correspond to the length and height of the box.