MARSVECTORCALC6 3.4.020. My Notes A rectangular box with no top is to have a surface area of 64 m2. Find the dimensions (in m) that maximize its volume.

We would have

where " l " is  length, " w"  is width and "h" is height.

Step-by-step explanation:

Step 1

Remember that

         Surface area for a box with no top =

where " l " is  length, " w"  is width and "h" is height.

 Step 2.

Remember as well that

                              Volume of the box =

Step 3

 We can now use lagrange multipliers.  Lets say,

and

By the lagrange multipliers method we know that                            

Step 4

Remember that

and

So basically you will have the system of equations

Now, remember that you can multiply the first eqation, by "l" the second equation by "w" and the third one by "h" and you would get

Then you would get

You can get rid of from these equations and you would get

And from those equations you would get

Now remember the original equation

If we plug in what we just got, we would have

127459073968947726

step-by-step explanation:

your welcome : )