Lesson Minimize surface area of a conical paper cup with the given volume

Minimize surface area of a conical paper cup with the given volume

Problem 1

The volume of the cone cup is given 40 cm^3.


The lateral area of a cone is

    S = ,       (1)

where r is the base radius and "s" is the slant height:  s = .

So, we need minimize the lateral area

    S =     (2)


at given restriction for the volume

     = 40  cm^3.    (3)


From the restriction (3),   

    h = . 


We substitute it into expression (2), and we get S(r) as a function of the radius r, only

    S(r) =  = 


    +-------------------------------------------------------+
    |    To find the minimum of S(r), we should calculate   |
    |         the derivative and equate it to zero.         |
    +-------------------------------------------------------+


I will not calculate the derivative in full, which is a complicated fraction. 
It is enough to calculate its numerator and equate it to zero. It gives this equation

    - +  = 0,

or, equivalently

    7200 = ,    = ,   r =  = 3.0 cm.    (4)


So, the radius is just found.  The height should be  

    h =  =  = 4.24 cm.    (5)


Expressions (4) and (5) give the required exact formulas and approximate numerical values.