a cylindrical can, open at the top, is to hold 800 cm cubed of liquid. find the height, h, and the radius, r, that minimize the amount of material needed to manufacture the can

Answer 1

V = 800 cm³ = πr²h (This is the constraint that ties r and h)

Cost of can material is proportional to surface area of the can without the top:

C = k (2πrh + πr²) (This is the function to minimize)

Substitute h = 800/πr²

C = k(2πr*(800/πr²) + πr²) = k(1600/r + πr²)

dC/dt = k(-1600/r² + 2πr) = 0

solve for r and use V relation to solve for h (2nd derivative is >0, min)

Please consider a tutor. Take care.