Question

A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced.

Based on the above information, answer the following questions:

If for a tumbler, V is its volume, h the height and r the radius of the circular base, then:

1. Differentiate its volume with respect to radius of the base, where the surface area is constant. (2)
2. If the company wants to maximize the volume of each tumbler, then establish a relation between its height and the radius of the base. (2)

Answer 1

Let the fixed surface area be S.

Surface area (S): S = πr² + 2πrh (Area of base + curved surface)

Volume (V): V = πr²h

From the surface area equation, we can express height h in terms of r and S:

2πrh = S − πr²

h = `(S - pir^2)/(2pir)`

Now, substitute h into the Volume formula:

V = `pir^2 ((S - pir^2)/(2pir))`

= `(r(S - pir^2))/2`

= `(Sr - pir^3)/2`

i.

Now we differentiate V with respect to r (S constant):

`(dv)/(dr) = d/(dr) ((Sr)/2 - (pir^3)/2)`

`(dv)/(dr) = S/2 - (3pir^2)/2`

ii.

To find the maximum volume, we set the first derivative to zero:

`(dv)/(dr) = 0`

`S/2 - (3pir^2)/2 = 0`

S = 3πr²

Now, substitute the value of S back into the original surface area equation to find the relation between h and r.

3πr² = πr² + 2πrh

3πr² − πr² = 2πrh

2πr² = 2πrh

h = r

For maximum volume, the height of the tumbler must be equal to the radius of its base.