Question 538428
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Since the assignment of the variables to the dimensions of the rectangle is purely arbitrary, we can assume that the *[tex \Large y] dimension of the rectangle becomes the height dimension of the cylider.  Our first task is to develop a function V, representing the volume of the cylinder, as a function of *[tex \Large x].


First, we know that the perimeter of the rectangle is 40, so we can write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 2y\ =\ 40]


From which we can derive


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 20\ -\ x]


The volume of a cylinder is found by multiplying the height by the area of the circular base.  Since the rectangle will be rolled into a cylinder, the *[tex \Large x] dimension of the rectangle will become the circumference of the circular base, hence the radius of the circular base is given by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ \frac{x}{2\pi}]


Using the formula for the volume of a cylinder, i.e. *[tex \Large V\ =\ \pi{r}^2h], and substituting the facts derived so far, we get a function of *[tex \Large x] that represents the volume of the cylinder:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ \pi\left(\frac{x}{2\pi}\right)^2\left(20\ -\ x\right)]


A little Algebra music, Mr. Spear...


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ \frac{20x^2\ -\ x^3}{4\pi}]


Find the extrema:  Set the first derivitive equal to zero.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ \frac{40x\ -\ 3x^2}{4\pi}]


Said function having zeros at *[tex \Large 0], which we can discard because it is perforce a minimum, and *[tex \Large \frac{40}{3}].


Evaluate the second derivitive at *[tex \Large \frac{40}{3}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ \frac{40\ -\ 6x}{4\pi}]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{40\ -\ 6\left(\frac{40}{3}\right)}{4\pi}\ <\ 0]


Therefore the extremum at *[tex \Large x\ =\ \frac{40}{3}] is a maximum.


Plug *[tex \Large x\ =\ \frac{40}{3}] back into *[tex \Large y\ =\ 20\ -\ x] to determine the *[tex \Large y] dimension.


<i><b>Super-deluxe Double-plus Extra Credit</b></i>


Derive the ratio between the circumference of the circular base and the height of a cylinder of maximum volume for a given cylinder height.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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