Question 1196998
<pre><font face = "consolas" color = "indigo" size = 4><b>
Hi  
A cone-shaped paper drinking cup is to hold  40cm^3 of water. 
Find the height and radius of the cup that will require the least amount of paper.
NOTE: Enter the exact answers. 
*[illustration cone]
V= 1/3 πr^2h = 40cm^3 0r {{{ h = 120/(pi*r^2)}}}
As my distinguished colleague pointed out... 
following REPRESENTS surface area of the 'cup'.
S = {{{ pi*r*sqrt(r^2 + (120/(pi*r^2))^2)}}}
Using graphing calculator: S' = ~0(.00003)  at r = ~3.0035cm
 r = ~3.0035  then h = {{{ h = 120/(pi*3.0035^2)}}} =4.2342
checking:
V= 1/3 πr^2h = 40cm^3
V= 1/3 π(3.0035^2)4.2342 = 39.9996 
AS Instructed:  "NOTE: Enter the 'exact answers. " 
Yes, one would need to use set S' = 0 & solve for r in terms of pi
That having been said:
The Graphing Calculator is a 'great way' to find answer to 4 decimal points.
Wish You the Best in your Studies.
</font></b></pre>