Mathematics
Mathematics
Problem
A cylindrical can, closed at both ends, must have a volume of . (i) Derive the radius and height that minimise the total surface area of the can. (ii) Use calculus to verify that the surface area obtained is the least possible.Intuitive explanation
This is an optimization problem where you're balancing two competing constraints: you need a fixed volume, but you want to minimize the material (surface area) used. The key insight is to express surface area as a function of just one variable by using the volume constraint to eliminate the other, then find where the derivative equals zero. That critical point will be a minimum because the surface area grows without bound as the radius or height becomes very small or very large.
Intuitive explanation
This is an optimization problem where you're balancing two competing constraints: you need a fixed volume, but you want to minimize the material (surface area) used. The key insight is to express surface area as a function of just one variable by using the volume constraint to eliminate the other, then find where the derivative equals zero. That critical point will be a minimum because the surface area grows without bound as the radius or height becomes very small or very large.