Question 133749
A cone-shaped paper cup is to hold 100 cubic cm of water. Find the height and the radius of the cup that can be made from the least amount of paper.
:
Use the volume of a cone formula: {{{1/3}}}*pi*r^2*h = V; to find h in terms of r.
{{{1/3}}}*pi*r^2*h = 100
multiply equation 3 to get rid of the denominator
pi*r^2*h = 300
h = {{{300/((pi*r^2))}}}
:
Using the surface area equation: SA = pi*r^2 + (pi*r*L), find L using r and h
L = {{{sqrt(h^2 + r^2)}}}
Substitute {{{300/((pi*r^2))}}} for h
L = {{{sqrt((300/((pi*r^2))) + r^2)}}}
Substitute above for L in the SA equation
:
SA = ({{{pi*r^2}}}) + ({{{pi*r}}}*{{{sqrt((300/((pi*r^2))) + r^2)}}})
:
Using this equation in my TI83, the graph showed the minimum radius to occur at appox 2.37 cm
:
Find the height using this value:
h = {{{300/((pi*2.37^2))}}}
h = 17 cm
;
:
Check solution by finding the volume
V = {{{1/3}}} * pi * 2.37^2 * 17
V = 99.994 ~ 100 cm
:
I was unable to produce the graph here. Hope this helps you.