Question

Find the dimensions of a right-circular cylinder that is open on the top and closed on the bottom, so that the can holds 1 liter and uses the least amount of material? ...?

Answer 1

Volume of cylinder:
V = πr²h
The desired volume is 1 Liter = 1000 cm³
1000 = πr²h
h = 1000/πr²

Surface area of cylinder:
S.A = 2πr² + 2πr²h
We substitute the value of h from the first equation:
S.A = 2πr² + 2πr(1/πr²)
S.A = 2πr² + 2/r
Now, to minimize surface area, we differentiate the expression with respect to r and equate to 0.
0 = 4πr - 1000/r²
4πr³ - 1000 = 0
r = 4.3 cm
h = 17.2 cm