Q. 18

# 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 :

Given,

• The can is cylindrical with a circular base

• The volume of the cylinder is 1 litre = 1000 cm^{2}.

• 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 = 1000cm^{3}

The Volume of the cylinder= πr^{2}h

1000 = πr^{2}h

---- (1)

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

S = πr^{2} + πr^{2} + 2πrh

S = 2 πr^{2} + 2πrh

[substituting (1) in the volume formula]

------ (2)

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:

[Since and ]

------- (3)

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:

----- (4)

[Since and ]

Now let us find the value of

As , so the function S is minimum at

Now substituting r in equation (1)

Therefore the radius of base of the cylinder, and height of the cylinder, where the surface area of the cylinder is minimum.

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