Question

A cylindrical can is to be made to hold 1 litre of oil. Find the dimensions which will minimize the cost of the metal to make the can.

Answer 1

Given,

• The can is cylindrical with a circular base

• The volume of the cylinder is 1 litre = 1000 cm².

• The surface area of the box is minimum as we need to find the minimum dimensions.

Let us consider,

• The radius base and top of the cylinder be ‘r’ units. (skin coloured in the figure)

• The height of the cylinder be ‘h’units.

• As the Volume of cylinder is given, V = 1000cm³

The Volume of the cylinder= πr²h

1000 = πr²h

h = 1000/πr² ---- (1)

The Surface area cylinder is = area of the circular base + area of the circular top + area of the cylinder

S = πr² + πr² + 2πrh

S = 2 πr² + 2πrh

[substituting (1) in the volume formula]

For finding the maximum/ minimum of given function, we can find it by differentiating it with r and then equating it to zero. This is because if the function f(r) has a maximum/minimum at a point c then f’(c) = 0.

Differentiating the equation (2) with respect to r:

To find the critical point, we need to equate equation (3) to zero.

Now to check if this critical point will determine the minimum surface area of the box, we need to check with second differential which needs to be positive.

Consider differentiating the equation (3) with r:

Now let us find the value of

so the function S is minimum at r = \(\sqrt[3]{\frac{500}{\pi}}\)

Now substituting r in equation (1)

Therefore the radius of base of the cylinder, r = \(\sqrt[3]{\frac{500}{\pi}}\) and height of the cylinder, h = \(\cfrac{1000}{\pi^{1/3}(500)^{2/3}}\) where the surface area of the cylinder is minimum.