13 of 20 QID: 26864 What is the radius of a right circular cylinder with a volume of 12 in3 if it has a minimum surface area

r = 1,248 in

Step-by-step explanation:

v(c) = 12 in³

The surface area of a right cylinder is:

Area of the base and top  + lateral area

S(a)  =  2*π*r²  + 2*π*r*h   (1)

v(c) = 12 in³  =  π*r²*h        h is the height of the cylinder, then

h  =  12 / π*r²

By substitution,  in equation (1) we get  the  Surface area as a function of r

S(r)  =  2*π*r² +   2*π*r*  ( 12 / π*r²)

S(r)  =  2*π*r² + 24 /r

Tacking derivatives on both sides of the equation we get:

S´(r)  =  4*π*r -  24 /r²

S´(r)  = 0            4*π*r -  24 /r²  =  0       π*r - 6/r² = 0

 π*r³ - 6  = 0

r³   =  1,91    

r = 1,248 in    

How do we know that the value  r  =  1,248 makes  Surface area minimum??

We get the second derivative

S´´(r)  =  4*π  +  48/r³    S´´(r)  will be always positive therefore we have a minumum of S   at the value of   r = 1,248 in