Question 265854
Tony needs a glass bell jar in the shape of a cylinder with a hemisphere on top. The height of the cylinder must be 3 inches longer than its radius and the volume must be 72pi cubic inches. What should the radius of the cylinder be?
:
Let r = the radius of the cylinder (and the hemisphere)
then
(r+3) = the height of the cylinder
:
vol of a sphere ={{{(4/3)pi*r^3}}}, therefore
vol of a hemisphere = {{{(2/3)pi*r^3}}}
:
vol of cylinder + vol of hemisphere = 72*pi
{{{pi*r^2*h}}} + {{{(2/3)pi*r^3}}} = {{{72pi}}}
:
Divide thru by pi and eliminate that
{{{r^2*h}}} + {{{(2/3)(r^3)}}} = 72
:
Replace h with (r+3) 
{{{r^2*(r+3)}}} + {{{(2/3)r^3}}} = 72
:
{{{r^3+3r^2)}}} + {{{(2/3)r^3}}} = 72
:
Add like terms r^3 + 2/3*r^3
{{{(5/3)(r^3)}}} + {{{3r^2}}} = 72
:
Multiply each term by 3
{{{5r^3 + 9r^2}}} = 216
:
{{{5r^3 + 9r^2 - 216}}} = 0
:
Graph this equation, find x intercept
{{{ graph( 300, 200, -4, 10, -100, 300, 5x^3+9x^2-216) }}}
r = 3 inches the radius
:
Check; find the volumes
{{{pi*3^2*6}}} = 169.646
{{{(2/3)pi*3^3}}} = 56.487
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Total volume is 226.195 divide this by pi you have 71.999 ~ 72