SOLUTION: a closed cylindrical tin is required to contain 500cc show that an equation for the total external surface area of the tin is area= 1000/r + 2pir^2 find the height and the d

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Question 1033470: a closed cylindrical tin is required to contain 500cc
show that an equation for the total external surface area of the tin is area= 1000/r + 2pir^2
find the height and the diameter of the can so as to use the minimum amount of metal?
Answer by Cromlix(4381) (Show Source):
You can put this solution on YOUR website!
Hi there,
π x r^2 x h = 500cc
h = 500/πr^2
Surface Area:
π x diameter x height + 2 x π x r^2
Using:
h = 500/πr^2
π x 2r (diameter)x 500/πr^2
Area = 1000/r + 2πr^2
Area = 1000r^-1 + 2πr^2
Area (differentiated)
= -1000r^-2 + 4πr
-1000r^-2 + 4πr = 0
-1000/r^2 + 4πr = 0
-1000/r^2 = -4πr (Multiply both sides by -1)
1000/r^2 = 4πr
1000 = 4πr^3
Divide both sides by 4
250 = πr^3
r^3 = 250/π
r = 3√250/π
r = 4.3
..........
Using Nature Table: - 4.3 +
.................. .......... ......... \ - /
Minimum.
...............
To establish height and radius
πr^2h = 500
π(4.3)^2h = 500
height = 500/(π(4.3)^2
Height = 8.6 centimeters
Radius = 4.3 centimeters
so, Diameter = 8.6 diameters.
Hope this helps :-)