Question 935948
If a cone has a volume of 500cm^3.
 What is it's radius and height with the smallest surface area.
:
Find the relationship between the height and radius using the volume
{{{(pi*r^2*h)/3 = 500}}}
multiply both sides by 3
{{{pi*r^2*h = 1500}}}
divide both sides by {{{pi*r^2}}}
{{{h = 1500/(pi*r^2)}}}
:
Surface area formula:
S.A = {{{pi*r*sqrt(r^2+h^2)}}}
replace h with {{{1500/(pi*r^2)}}}
S.A. = {{{pi*r*sqrt(r^2 + (1500/(pi*r^2))^2)}}}
Graph this equation
{{{ graph( 300, 200, -4, 15, -200, 500, pi*x*sqrt(x^2 + (1500/(pi*x^2))^2) ) }}}
minimum surface area occurs when r = 7 cm, about 264 sq cm
Find the height
h = {{{1500/(pi*7^2)}}}
h = 9.7 cm
:
See if that checks out by finding the vol with these value
v = {{{1/3}}}*{{{pi*7^2*9.7}}}
v = 498 cu/cm, is that close enough?