A closed box has a square base with side length feet and height feet. Given that the volume of the box is cubic feet, express the surface area of the box in terms of only.

Complete Question:

A closed box has a square base with side length L feet and height h feet. Given that the volume of the box is 39 cubic feet, express the surface area of the box in terms of L only.

Step-by-step explanation:

Given

Required

Express the surface area in terms of L

Because the box has a square base:

The volume is:

Where

So, we have:

Substitute 39 for Volume

Make H the subject

The surface area (A) of a box with square base is:

Open bracket

Substitute for H

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.