Q. 205.0( 1 Vote )

# Find the radius o

Answer :

Given,

• The closed is cylindrical can with a circular base and top.

• The volume of the cylinder is 1 litre = 100 cm^{3}.

• The surface area of the box is minimum.

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 = 100cm^{3}

The Volume of the cylinder= πr^{2}h

100 = 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.

---- (4)

__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:

----- (5)

[Since and ]

Now let us find the value of

As , so the function S is minimum at

As S is minimum from equation (4)

V = 2πr^{3}

Now in equation (1) we have,

h = 2r = diameter

Therefore when the total surface area of a cone is minimum, then height of the cone is equal to twice the radius or equal to its diameter.

Rate this question :

If the sum of theMathematics - Board Papers

A metal box with Mathematics - Board Papers

Show that aRD Sharma - Volume 1

Find the local maMathematics - Board Papers

Prove that the seMathematics - Board Papers

Prove that the raMathematics - Board Papers

Prove that the leMathematics - Board Papers