SOLUTION: A tank is made up of a cyliner 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 t

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Question 390380: A tank is made up of a cyliner 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 cacacity of the 2hemispheres combined is 5:4. Calculate the internal radius length?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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
%28%284%2F3%29pi%2Ar%5E3%29+%2B+%28pi%2Ar%5E2%2Ah%29 = 81%2Api cu/cm
factor out pi
%28%284%2F3%29%2Ar%5E3%29+%2B+%28r%5E2%2Ah%29 = 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."
%283r%5E2h%29%2F%284r%5E3%29 = 5%2F4
cross multiply
4(3r^2h) = 5(4r^3)
12r^2h = 20r^3
divide both sides by r^2
12h = 20r
h = %2820r%29%2F12 = %285r%29%2F3
:
4r^3 + 3r^2*h = 243
replace h
4r^3 + 3r^2*%285r%29%2F3 = 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=%285%2A3%29%2F3
h = 5 cm is the height
%28%284%2F3%29%2A3%5E3%29+%2B+%283%5E2%2A5%29 = 81
36 + 45 = 81; confirms our solution of r = 3