If 30,000 cm2 of material is available to make a box with a square base and an open top, what is the largest possible volume (in cm3) of the box?

Question

asked Oct 25, 2022 107k views

1 Answer

Wako · Answer 1 · 2022-10-31T07:04:42+0000

Answer:

The largest possible volume of the box is 2000000 cubic meters.

Explanation:

The volume (
), in cubic centimeters, and surface area (
A_(s) ), in square centimeters, of the box with a square base are described below:

$A_(s) = l^(2)+h\cdot l$ (1)

$V = l^(2)\cdot h$ (2)

Where:

- Side length of the base, in centimeters.

- Height of the box, in centimeters.

By (2), we clear
within the formula:

h = (V)/(l^(2))

And we apply in (1) and simplify the resulting expression:

A_(s) = l^(2)+ (V)/(l)

$A_(s)\cdot l = l^(3)+V$

$V = A_(s)\cdot l -l^(3)$ (3)

Then, we find the first and second derivatives of this expression:

$V' = A_(s)-3\cdot l^(2)$ (4)

$V'' = -6\cdot l$ (5)

If
V' = 0 and
$A_(s) = 30000\,cm^(2)$ , then we find the critical value of the side length of the base is:

$30000-3\cdot l^(2) = 0$

$3\cdot l^(2) = 30000$

$l = 100\,cm$

Then, we evaluate this result in the expression of the second derivative:

V'' = -600

By Second Derivative Test, we conclude that critical value leads to an absolute maximum. The maximum possible volume of the box is:

$V = 30000\cdot l - l^(3)$

$V = 2000000\,cm^(3)$

The largest possible volume of the box is 2000000 cubic meters.