Question

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.

Answer 1

Answer:

We would have

                                    $l =w =\frac{8\sqrt{3}}{3} \\h = \frac{8\sqrt{3}}{6}$

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 = $lw+2lh+2wh = 64$

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

 Step 2.

Remember as well that

                              Volume of the box = $l*w*h$

Step 3

 We can now use lagrange multipliers.  Lets say,

                                    $F(l,w,h) = lwh$

and

                                $g(l,w,h) = lw+2lh+2wh = 64$

By the lagrange multipliers method we know that                            

 

                                                     $\nabla F = \lambda \nabla g$

Step 4

Remember that

                          $\nabla F = (wh,lh,lw)$

and

                      $\nabla g = (w+2h,l+2h , 2w+2l)$

So basically you will have the system of equations

                              $wh = \lambda (w+2h)\\lh = \lambda (l+2h)\\lw = \lambda (2w+2l)$

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

                                   $lwh = l\lambda (w+2h)\\\\lwh = w\lambda (l+2h)\\\\lwh = h\lambda (2w+2l)$

Then you would get

                      $l\lambda (w+2h) = w\lambda (l+2h) = h\lambda (2w+2l)$

You can get rid of $\lambda$ from these equations and you would get

                         $lw+2lh = lw+2wh = 2wh+2lh$

And from those equations you would get

                                             $l = w =2h$

Now remember the original equation

                                    $lw+2lh+2wh = 64$

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

                                 $l^{2} + l^{2} + l^2 = 64 \\3l^{2} = 64 \\l = w = \frac{8\sqrt{3} }{3} \\h = \frac{8\sqrt{3} }{6}$