find the dimensions of the can (radius and height) that will minimize the amount of material used (surface area)

Question

a cylindrical soda can is being designed to hold a volume of approximately 23.56 in3

Answer 1

V = (pi)r²h  and surface area = 2(pi)r² + 2(pi)rh.  Since we know the volume we can solve for h in terms of r

 

23.56 = (pi)r²h

 

h = 23.56/((pi)r²) and substitute this fin the second equation

 

SA = 2(pi)r² + 2(pi)r(23.56/((pi)r²)  and simplify

 

SA = 2(pi)r² + 47.12r^-1  and you need to find a relative minimum, differentiate and find the critical values

 

4(pi)r - 47.12r^-2 = 0

 

4(pi)r = 47.12/r²

 

r³ = 47.12/4(pi)

 

r = 1.554 inches  then solve for the height from above

 

23.56 = (pi)(1.554)²h

 

h = 3.105 inches