Finding the cylinder radius that minimizes surface area for a hemisphere-capped cylinder

[calvinnn]

optimization problem!

OKOK running out of time! CAn anyone please help me with this problem:

Surface Area A solid os formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radiusof the cylinder that produces the minimum surface area.

if anyone can help me with this, ill be VERY grateful!

 

[Tide]

Exactly what have you tried so far?

 

[HallsofIvy]

1. Call the length of the cylinder l and the radius r. Write down the formula for the volume of the figure (its a cylinder and a sphere) and set that equal to 12. Solve for l.

2. Write down the formula for surface area (again, lateral area of a cylinder, area of a sphere) and replace l by the formula from 1 so that you have a function of r only.

3. Find the value of r that minimizes that function.

 

[Sick0Fant]

Q: Surface Area A solid os formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radiusof the cylinder that produces the minimum surface area.

A:

SA=2 π r2 + 2 π r h

V=πr^2h, V=12

h=12/(πr^2)

thus, SA=(2πr^2)+(2πr)(12/(πr^2))

SA'=4πr-(24/r^2) To find the minimum, set SA' to zero.

0=4πr-(24/r^2) r=6^(1/3)/π^(1/3), approx. 1.2

Test a point on either side of r=1.2 to make sure that it is a minimum.