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Calculus/calculus Optimization Problem


Postal regulations specify that a package sent by a certain shipping method may have a combined length and girth of no more than 99 inches. Find the dimensions of a cylindrical package with the greatest volume that may be sent by this shipping method. Hint: the length plus the girth is 2pir+l r=radius l=length

I never use the letter l unless I capitalize it if it can be helped.
This is because l looks to much like the number 1.
Let us take the length to be called the height so I can use h instead of l.

It is known that 99 = h + 2r, so h = 99 - 2r.

Since volume is V = πrh, and h = 99 - 2r, this gives us V(r) = 99πr - 2πr.
That is that same as V(r) = π(99r - 2r).

From here, it is fairly easy to see that V'(r) = π(198r - 6r).
Since 198 = 6 * 33, that can be factored into V'(r) =  6πr(33 - r).

Solving gives us 6πr = 0 and 33 - r = 0, from which we get r = 0 and r = 33.

Since the value of r = 0 gives us a minimum volume, and the volume is a cubic in r, that means r = 33 must be a maximum.  This is because it can be seen that cubic equations always have a maximum and a minimum.  Another way to see that it is a maximum is to take the 2nd derivative, put r = 33 in, and getting a negative value { V"(r) =  π(198 - 12r), 12*33 = 396,
198 - 396 = -198 }.

Since h = 99 - 2r, it can be seen that this gives h = 99 - 66 = 33.  


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