Ole D.

asked • 05/19/20

largest volume of cylinder

A rectangle has a circumference of 36 cm and sides x cm and y cm. It should be rolled to a height y cylinder.

Determine x and y so that the cylinder gets the largest volume possible.


Stanton D.

Hi Ole D., suggest you express "y" in terms of "x", and write the expression for the volume of the cylinder. Then take the first derivative of that volume expression with respect to x, set that = 0, and solve! -- Cheers, - - Mr. d.
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05/20/20

1 Expert Answer

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Kathy P. answered • 05/20/20

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C = 2x + 2y

36 = 2x + 2y

18 = x + y ==>. y = 18-x. This is the height of cylinder


Find radius of cylinder, when rectangle is rolled into a cylinder.

Circumference of cylinder is x and the height is y

C = 2*pi*r

x = 2*pi*r

r = x/(2*pi)


Vol = Base Area * Height

Vol = (pi*r^2) * Height

Vol = pi*[x/(2*pi)] * (18-x). Now, Volume is an equation in terms of x

Vol = (x/2)*(18-x). Zeros when x = 0, 18


The vertex is between the two zeroes.

The parabola is concave down so the vertex is a max.

The max occurs when x = 18/2 = 9


Max. Vol = (9/2)*(18-9)

Max. Vol = (9/2)*(9)

Max. V0l = 81/2 units^3



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Kathy P.

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x = 9 cm y = 18-x = 18-9 = 9 cm max. Vol = 81/2 cm^3
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05/20/20

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