Question

A manufacturer needs to make a cylinder can that will hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in itd construction.

Answer 1

Determine the dimensions of the cylinder given a volume of 1.5 liters.
so,
1.5 liters = formula for a cylinder volume
1.5 liters = pi*r^2 (h)


minimize the amount of material used
so,
material used is equivalent to the surface area of the cylinder.
area = (2)pi*r^2 + (h) 2pi*r

Answer 2

we need to convert liters meters^3
1.5L = 0.0015 m^3
0.0015 = pi*r^2 (h)
set equal to h
h = 0.0015 / (pi*r^2)

combine with area equation by replacing h
area = (2)pi*r^2 + (h) 2pi*r
area = (2)pi*r^2 + 2pi*r (0.0015 / (pi*r^2))
to 'minimize' we need to take the derivative of area with respect to r, set it equal to zero. solve for r. once you have r, plug it into 0.0015 = pi*r^2 (h) and solve for h