Answer: Radius = 32.7 cm and height = 65.5 cm
Explanation:
An oil drum must have a volume of 218 liters or 218*1000cm^3 = 218,000 cm^3
The price is per square meter, so if we reduce the surface of the oil drum, we will pay less:
So we want to play with the measures of the oil drum in such a way that the surface is minimized.
Now, first, the volume of the oil drum (a cylinder) is:
V = pi*r^2*h
where pi = 3.14, r is the radius and h is the height.
and the surface is:
S = 2*pi*r^2 + 2*pi*r*h
we know that:
pi*r^2*h = 218,000 cm^3
r^2*h = 218,000cm^3/3.14 = 70,096.5 cm^3
now we can write
h = 70,096.5 cm^3/r^2
now we can replace it in the surface equation:
S = 2*pi*r^2 + 2*pi*r*h
= 6.28*(r^2 + 70,096.5 cm^3/r)
So we want to minimize this, we can derivate it and find the zero:
S' = 6.28(2*r - 70,096.5 cm^3/r^2) = 0
2r = 70,096.5 cm^3/r^2
r^3 = (70,096.5 cm^3)/2
r = ∛( (70,096.5 cm^3)/2 ) = 32.7cm
And then the height is:
h = 70,096.5 cm^3/r^2 = 70,096.5 cm^3/(32.7cm)^2 = 65.5cm
