Question

Read the following passage:

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. One complete of a four-cylinder four-stroke engine. The volume displace is marked The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π cm2.

Based on the above information, answer the following questions:

1. If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
2. Find `(dV)/(dr)`. (1)
3. (a) Find the radius of cylinder when its volume is maximum. (2)
   OR
   (b) For maximum volume, h > r. State true or false and justify. (2)

Answer 1

i. Area of metal sheet required to made a cylinder open from top = 75π cm².

Given, 'r' is the radius of the cylinder 'h' is the height of the cylinder.

∴ 2πrh + πr² = 75π

`\implies` 2rh + r² = 75

`\implies` 2rh = 75 – r²

`\implies` h = `(75 - r^2)/(2r)`  ...(i)

Then, volume of cylinder,
ii. From (i),

V = πr²h

= `πr^2 xx ((75 - r^2)/(2r))` ...[From (i)]

= `(πr)/2 (75 - r^2) cm^3`.

V = `π/2 (75r - r^3)`

`(dV)/(dr) = π/2 (75 - 3r^2)`   ...(ii)

iii. (a) From (ii),

`(dV)/(dr) = π/2 (75 - 3r^2)`

For maximum volume, put `(dV)/(dr)` = 0

`\implies π/2(75 - 3r^2)` = 0

`\implies (3pi)/2 != 0 , 25-r^2 = 0`

`\implies` 25 = r²  [after taking square roots]

`\implies` r = 5 cm

Now, `(d^2V)/(dr^2) = pi/2(– 3r^2) < 0`

Hence, the volume is maximum, when r = 5 cm.

OR

(b) [From ...(i)]

Then, h = `(75 - r^2)/(2r)` 

`= (75-5^2)/(2(5))`

= `(75 - 25)/(10)`

= `50/10`

= 5

For max. volume,

h = r = 5 cm

Hence, the given statement is false.