Jack unfolded a cardboard box. the figure of the unfolded box is shown below:

which expression can be used to calculate the area of cardboard, in square inches, that was used to make the box?

8 x 6 x 6
6 x 4 x 4
4 x 6 x 6
6 x 8 x 8

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.