Design a Cylinder

Question

Design a cylinder can of volume 1000cm^3 so that it uses the least amount of metal. Show that h=2r, where h is the height of the cylinder and r is its radius.

Answer 1

The surface area of the cylindrical can consists of the areas of two circular bases as

well as the rectangle formed by "peeling off the label"... that is the height of the cylinder

times the circumference of the circular base.

Surface Area SA = 2*pi*r * h + 2 * pi * r^2

is the function to be minimized

We have

V = 1000 = pi*r^2 * h

1000/(pi*r^2) = h

Substituting into Surface area formula

SA = 2 * pi * r * 1000 / (pi * r^2) + 2 * pi * r^2

= 2000/r + 2 * pi * r^2

0 = dSA/dR = -2000*r^(-2) + 4 * pi * r

Multiplies by r^2

0 = -2000 + 4 * pi * r^3

2000/(4 *pi) = r^3

500/pi = r^3

r = 5.4192607031929-

which forces h=10.8385214-