Optimizing Volume: Solving the Maximum Value Problem with Critical Point Formula

[Dissonance in E]

Homework Statement

find the largest volume of a rectangular box that satisfies the following condition
the sum of the height and horizontal perimeter does not exceed L

Homework Equations

critical point formula:
system of equations must satisfy the following at critical values of x & y
fx = 0
fy = 0

The Attempt at a Solution

height + (width * depth) = L
height*width*depth = V

I do know the answer to be L^3/108 cubic units, but as to how to get that is beyond me.

 

[rl.bhat]

Homework Statement

find the largest volume of a rectangular box that satisfies the following condition
the sum of the height and horizontal perimeter does not exceed L

Homework Equations

critical point formula:
system of equations must satisfy the following at critical values of x & y
fx = 0
fy = 0

The Attempt at a Solution

height + (width * depth) = L
height*width*depth = V

I do know the answer to be L^3/108 cubic units, but as to how to get that is beyond me.

Horizontal perimeter = 2(x+y) where x is the width and y is the depth.

Area of the base is maximum when x = y. To have maximum volume, change h so that ( h + 2x + 2y) = L.

If you put x = y, Volume V = h*x^2

Put x = (L-h)/4. Find dV/dh and equate it to zero. Find h in terms of L and find V.

 

[Dissonance in E]

ahh i got it now, thanks a bunch!