SOLUTION: A cylindrical petrol storage tank is to be constructed of metal on a concrete base. Find the relation between the radius of the base and the height of a tank so that the amount o

Algebra ->  Matrices-and-determiminant -> SOLUTION: A cylindrical petrol storage tank is to be constructed of metal on a concrete base. Find the relation between the radius of the base and the height of a tank so that the amount o      Log On


   

Question 875784: A cylindrical petrol storage tank is to be constructed of metal on a concrete
base. Find the relation between the radius of the base and the height of a
tank so that the amount of metal used is the least possible to hold a given
volume V0.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
For a cylinder,
V%5B0%5D=pi%2AR%5E2%2AH
For the surface area of the tank (amount of metal used to make it),
A=2%2Api%2AR%5E2%2B2%2Api%2AR%2AH
From above,
H=V%5B0%5D%2F%28pi%2AR%5E2%29
Substituting,
A=2%2Api%2AR%5E2%2B2%2Api%2AR%2A%28V%5B0%5D%2F%28pi%2AR%5E2%29%29
A=2%2Api%2AR%5E2%2B%282V%5B0%5D%29%2FR
To find the minimum amount of metal take the derivative of A and set it to zero,
dA%2FdR=4%2Api%2AR-%282V%5B0%5D%29%2FR%5E2=0
4%2Api%2AR=%282V%5B0%5D%29%2FR%5E2
V%5B0%5D=2pi%2AR%5E3
From above,
pi%2AR%5E2%2AH=2pi%2AR%5E3
H=2R