(D) The volume of a cylinder is given by $V = \pi r^2 h$.Differentiating both sides with respect to time $t$,we get:$\frac{dV}{dt} = \pi \left( 2rh \f

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$33 \pi \text{ cm}^3/\text{sec}$

(D) The volume of a cylinder is given by $V = \pi r^2 h$.Differentiating both sides with respect to time $t$,we get:$\frac{dV}{dt} = \pi \left( 2rh \frac{dr}{dt} + r^2 \frac{dh}{dt} \right)$Given: $r = 3 \text{ cm}$,$h = 5 \text{ cm}$,$\frac{dr}{dt} = 2 \text{ cm/sec}$,and $\frac{dh}{dt} = -3 \text{ cm/sec}$ (since height is decreasing).Substituting these values into the derivative:$\frac{dV}{dt} = \pi \left( 2 \times 3 \times 5 \times 2 + 3^2 \times (-3) \right)$$\frac{dV}{dt} = \pi \left( 60 - 27 \right)$$\frac{dV}{dt} = 33 \pi \text{ cm}^3/\text{sec}$.

Class 12 Mathematics Applications of Derivatives Rate of Change of Quantities

Medium

The radius of a cylinder is increasing at the rate of $2 \text{ cm/sec}$ and its height is decreasing at the rate of $3 \text{ cm/sec}$. Find the rate of change of volume when the radius is $3 \text{ cm}$ and the height is $5 \text{ cm}$.

MHT CET 2022 All MHT CET

A
$44 \pi \text{ cm}^3/\text{sec}$
B
$11 \pi \text{ cm}^3/\text{sec}$
C
$23 \pi \text{ cm}^3/\text{sec}$
D
$33 \pi \text{ cm}^3/\text{sec}$

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The radius of a cylinder is increasing at the rate of $2 \text{ cm/sec}$ and its height is decreasing at the rate of $3 \text{ cm/sec}$. Find the rate of change of volume when the radius is $3 \text{ cm}$ and the height is $5 \text{ cm}$.

Similar Questions

$A$ closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After $21 \, min$,it was found that the height of the water column is half of the original height. How much more time in minutes does it require to empty the vessel?

The radius of a right circular cylinder increases at the rate of $0.1 \text{ cm/min}$ and the height decreases at the rate of $0.2 \text{ cm/min}$. The rate of change of the volume of the cylinder in $\text{cm}^3\text{/min}$,when the radius is $2 \text{ cm}$ and the height is $3 \text{ cm}$,is:

The ordinate of a point describing the circle $x^2 + y^2 = 25$ decreases at the rate of $1 \, cm/sec$. Find the rate of change of the abscissa of the point when the ordinate is equal to $3 \, cm$ (Given $x > 0, y > 0$).

$A$ point moves along the arc of the parabola $y = 2x^2$. Its abscissa increases uniformly at the rate of $2 \text{ units/sec}$. At the instant the point is passing through $(1, 2)$,its distance from the origin is increasing at the rate of

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