SOLUTION: What is the maximum volume of a open box created from a square of side length 24? Keep in mind that V=LWH. So for this problem the dimensions are as follows:
L = 24-2x, W = 24-2x
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Question 664592: What is the maximum volume of a open box created from a square of side length 24? Keep in mind that V=LWH. So for this problem the dimensions are as follows:
L = 24-2x, W = 24-2x and H = x. The maximum volume will be where the volume equation is the greatest.
You are to determine:
1. The volume equation.
2. The maximum volume.
3. The value of x that maximizes the volume.
Explain each step.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Multiply length times width times height:
^2\left(x\right)\ =\ \left(4x^2\ -\ 96x\ +\ 576\right)\left(x\right)\ =\ 4x^3\ -\ 96x^2\ +\ 576x)
So
\ =\ 4x^3\ -\ 96x^2\ +\ 576x)
Take the first derivative (Power and Sum Rules)
Set the first derivative equal to zero and solve. This is a quadratic so you will get two roots, each of which represents the abscissa of a local extremum of the original function.
Take the second derivative:
Evaluate
 )
where 
is either of the abscissas of the possible extreme points. If  \ <\ 0)
then is the abscissa of a local maximum, if \ >\ 0)
then is the abscissa of a local minimum. If \ =\ 0)
, the second derivative test is inconclusive, but it is a possible inflection point. In the latter case, for this problem, calculate the volume for the value of in question and see if it makes sense as a possible 
value that could give a maximum volume.
John
My calculator said it, I believe it, that settles it
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