Posted by Ashley on Wednesday, November 7, 2012 at 3:13pm.
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 - Steve, Wednesday, November 7, 2012 at 3:24pm
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 - Ashley, Wednesday, November 7, 2012 at 3:30pm
Is pi in the denominator?
- Calculus - Steve, Wednesday, November 7, 2012 at 3:39pm
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