Question 538428: A rectangular sheet of perimeter 40 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y given the largest cylinder volume?
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! A rectangular sheet of perimeter 40 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y given the largest cylinder volume?
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2(x + y) = 40
y = 20 - x
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x = 2*pi*r
r = x/2pi
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Vol = pi*r^2*y = pi*(x/2pi)^2*(20 - x)
Vol = x^2*(20-x)/4pi
dV/dx = (1/4pi)*(40x - 3x^2) = 0
3x^2 = 40x
x = 40/3
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Local max at x = 40/3, y = 20/3 cm
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PS Max volume = 94.314 cc
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Since the assignment of the variables to the dimensions of the rectangle is purely arbitrary, we can assume that the  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  .
First, we know that the perimeter of the rectangle is 40, so we can write:
From which we can derive
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 dimension of the rectangle will become the circumference of the circular base, hence the radius of the circular base is given by:
Using the formula for the volume of a cylinder, i.e.  , and substituting the facts derived so far, we get a function of that represents the volume of the cylinder:
\ =\ \pi\left(\frac{x}{2\pi}\right)^2\left(20\ -\ x\right))
A little Algebra music, Mr. Spear...
\ =\ \frac{20x^2\ -\ x^3}{4\pi})
Find the extrema: Set the first derivitive equal to zero.
Said function having zeros at  , which we can discard because it is perforce a minimum, and  .
Evaluate the second derivitive at
and
}{4\pi}\ <\ 0)
Therefore the extremum at  is a maximum.
Plug back into  to determine the dimension.
Super-deluxe Double-plus Extra Credit
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
My calculator said it, I believe it, that settles it
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