I need help with this math question?

1 liter = 1000 cm³
Area of an open top cylinder = A = πr² + 2πrh
Volume = 1000 = πr²h
we have to eliminate one of the variables.
h = 1000/πr²
A = πr² + 2πrh
A = πr² + 2πr(1000/πr²)
A = πr² + 2000/r

to get min/max, differentiate and set equal to zero
dA/dr = 2πr – 2000/r² = 0
2000/r² = 2πr
πr³ = 1000
r = 6.83 cm
h = 1000/πr² = 6.83 cm

Let r = radius and h = height

Then

πr^2h = 1000 cm^3 ....(1).............. volume

S = πr^2 + 2πry ...........(2) .......... surface area

Rom (1)

h = 1000/πr^2 ...........(3)

Put this in (2)

S = πr^2 + 2πr( 1000/πr^2)

= πr^2 + 2000/r

dS/dr = 2πr - 2000/r^2

For least amount of metal dS/dr = 0

2πr - 2000/r^2 = 0

2πr^3 - 2000 = 0

r^3 = 2000/2π

r = ∛(1000/π)

= 6.827840633 cm .............you correct to required accuracy



Put this in (3)

h = 1000/π(6.827840633)^2

= 6.827840632 cm.............you correct to required accuracy