Final answer:
The width and length of the box are 6 feet, and the height is 11 feet. These dimensions meet both the given volume of 396 ft³ and the surface area of 300 ft², confirming that the correct answers are a) and c).
Step-by-step explanation:
To find the dimensions of the box, let us denote the width (and also the length, because the base is square) as 'x' feet. The height then would be 'x+5' feet, according to the problem. As the box is open-topped, the surface area (SA) consists of the square base and the four sides. The volume (V) of the box is the product of the base area and the height.
The formula for the volume of a box is V = length x width x height, so we have:
V = x * x * (x+5) = 396
x³ + 5x² = 396
The formula for the surface area of a box with an open top is SA = x² + 4 * x * height, so we have:
SA = x² + 4 * x * (x + 5) = 300
x² + 4x² + 20x = 300
5x² + 20x - 300 = 0
This can be simplified to:
x² + 4x - 60 = 0
Solving this quadratic equation, we find that the width (and length) of the box is x = 6 ft. Therefore, the height is x + 5 = 11 ft.
Checking these values against the volume:
V = 6 * 6 * 11 = 396
And the surface area:
SA = 6² + 4 * 6 * 11 = 300
Therefore, the correct answers are: a) The width (and length) of the box is 6 ft and c) The height of the box is 11 ft. This is a meeting of volume and surface area criterias which confirms our solution.