(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 Application 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 $5 \text{ cm}$.

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The volume of a cube is increasing at a rate of $9 \text{ cm}^3/\text{s}$. How fast is the surface area increasing when the length of an edge is $10 \text{ cm}$ (in $\text{ cm}^2/\text{s}$)?

If $f(x) = x^{2}$,find $\frac{f(1.1) - f(1)}{1.1 - 1}$.

$A$ ladder $5 \ m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground,away from the wall,at the rate of $2 \ cm/s$. How fast is its height on the wall decreasing when the foot of the ladder is $4 \ m$ away from the wall?

If the rate of increase of the area of a circle is not constant but the rate of increase of the perimeter is constant,then the rate of increase of the area varies:

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