Question

Consider a rectangle of perimeter 12 cm. form a cylinder by revolving this rectangle about one of its edges. what dimensions of the rectangle can give us a cylinder of maximum volume?

Answer 1

Dimensions will be 4 * 2 cm.

Explanation:

l = length of rectangle and w = width

Perimeter = 2l + 2w = 12

l + w = 6.

---> l = 6 - w

Volume of the cylinder

V = πr^2l

w = 2πr

--> r = w/2π

l = 6 - w so

V = π(w/2π)^ 2 * (6 - w)

---> V = w^2/4π (6- w)

---> V = 3w^2/ 2π - w^3/4π

Differentiating:

dV/ dw = 6w/ 2π - 3w^2 / 4π

= - 3(w - 4)w / 4π

This equals 0 for maximum volume

- 3(w - 4)w / 4π = 0

w = 0 or w = 4