Question 390380
A tank is made up of a cylinder with a hemisphere stuck to either side.
 The capacity (internal volume) of the tank is 81(pi) cm cubed.
 The ratio of the capacity of the cylinder to the capacity of the 2 hemispheres combined is 5:4.
 Calculate the internal radius length? 
:
Let r = the radius
:
The volume equation
:
2 hemispheres + cylinder = total volume 
{{{((4/3)pi*r^3) + (pi*r^2*h)}}} = {{{81*pi}}} cu/cm
factor out {{{pi}}}
{{{((4/3)*r^3) + (r^2*h)}}} = {{{81}}} cu/cm
get rid of the denominator, multiply by 3, results:
4r^3 + 3r^2*h = 3(81)
4r^3 + 3r^2*h = 243
:
"The ratio of the capacity of the cylinder to the capacity of the 2 hemispheres combined is 5:4."
{{{(3r^2h)/(4r^3)}}} = {{{5/4}}}
cross multiply
4(3r^2h) = 5(4r^3)
12r^2h = 20r^3
divide both sides by r^2
12h = 20r
h = {{{(20r)/12}}} = {{{(5r)/3}}}
:
4r^3 + 3r^2*h = 243
replace h
4r^3 + 3r^2*{{{(5r)/3}}} = 243
Cancel 3
4r^3 + r^2*5r = 243
4r^3 + 5r^3 = 243
9r^3 = 243
Divide both sides by 9
r^3 = 27
Find the cube root of 27
r = 3 cm is the radius
:
:
See if this is true
find h: h={{{(5*3)/3}}}
h = 5 cm is the height
{{{((4/3)*3^3) + (3^2*5)}}} = {{{81}}}
36 + 45 = 81; confirms our solution of r = 3