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प्रश्न
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.
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Based on the above information, answer the following questions:
- 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)
- Find `(dV)/(dr)`. (1)
- (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)
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उत्तर
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.
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