# Calculus

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A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 2 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.)

• Calculus -

if the cylinder is of radius r and height h, the volume v is

v = 4/3 pi r^3 + pi r^2 h

so, h = (v - 4/3 pi r^3)/(pi r^2)
h = (2 - 4/3 pi r^3)/(pi r^2)

the surface area a is

a = 4pi r^2 + 2pi r h
= 4pi r^2 + 2pi r (2 - 4/3 pi r^3)/(pi r^2)
= 4pi r^2 + 2/r (2 - 4/3 pi r^3)
= 4pi r^2 + 4/r - 8pi/3 r^2
= (4pi - 8pi/3) r^2 + 4/r
= 4pi/3 r^2 + 4/r

maximum area where da/dr = 0

da/dr = 8pi/3 r - 4/r^2
= (8pi/3 r^3 - 4)/r^2

da/dr=0 when r = ∛(3/2π)

As usual, check my math to verify result.

• Calculus -

Is pi in the denominator?

• Calculus -

yes.

• Calculus -

what is r?

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