Rate of Change of Quantities

A particle is moving in a straight line so that after $t$ seconds its distance $s$ (in centimetres) from a fixed point on the line is given by $s=f\left(t\right)=8t+t^3$. Find the initial velocity.

The volume of a cube is increasing at the rate of $18 {\mathrm{cm}}^3$ per second. When the edge of the cube is $\mathrm{12 }\mathrm{c}\mathrm{m,}$ then the rate in $\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ at which the surface area of the cube increases, is

Sand is pouring from a pipe at the rate of $12 {\mathrm{cm}}^3/\mathrm{s}$. The falling sand forms a cone on the ground in such a w that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand co increasing when the height is $4 \mathrm{cm}$?  [Report your answer upto the 4th decimal place]

The displacement is $\text{'}s\text{'}$ of the particle at time $\text{'}t\text{'}$ is given by $s=t^3-4t^2-52$. Find its velocity and acceleration at time $t=2s$.

The surface area of a spherical balloon is increasing at the rate of $2 {\mathrm{cm}}^2/\sec$. If the volume of the balloon is increasing at $a {\mathrm{cm}}^3/\sec$, when the radius of the balloon is $6 \mathrm{cm}$, then $a$ is

The surface area of a spherical balloon is increasing at the rate of $2 {\mathrm{cm}}^2/\sec$. At what rate $\left({\mathrm{cm}}^3/\sec \right)$ the volume of the balloon is increasing ,when the radius of the balloon is $6 \mathrm{cm}?$

Sand is pouring from a pipe at the rate of $12 {\mathrm{cm}}^3/\mathrm{s}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4 \mathrm{cm}?$ [Report your answer upto the 4th decimal place]

Water is being poured at the rate of $36{\mathrm{m}}^3/\sec$ in a cylindrical vessel of base radius $3$ meters. If the rate at which water level is rising is of the form $\frac{a}{\pi }$, then the value of $a=$

A square plate is contracting at the uniform rate of $2 c{m}^2/sec$. Find the rate of decrease of its perimeter when the side of the square is $16 cm$ long

For a gas equation $\mathrm{PV}=100$, volume $\mathrm{V}=25 {\mathrm{cm}}^3$ and pressure $\mathrm{P}$ is measured in $\mathrm{dynes}/{\mathrm{cm}}^2$. Find the rate of change of pressure when the volume is increasing at the rate of $0.25 c{m}^3/sec$

A ladder of $5 \mathrm{m}$ long rest with one end against a vertical wall of height $3 \mathrm{m}$ and the other end on the lower ground. If its top slides down at the rate of $10 cm/sec$, find the rate at which the foot of the ladder is sliding.

The area of the square is increasing at the rate of $0.5$ cm/sec. Find the rate of increase of its perimeter when the side of the square is $10$ cm long

The surface area of a spherical balloon is increasing at the rate of $2{\mathrm{cm}}^2/\sec$. At what rate is the volume of the balloon is increasing when the radius of the balloon is $6\mathrm{cm}$ ?

A stone is dropped into a pond. Waves in the form of circles are generated and the radius of the outermost ripple increases at the rate of $2$ inches/sec. Then, find the rate at which the area is increasing after $5$ seconds.

A stone is dropped into a pond. Waves in the form of circles are generated and the radius of the outermost ripple increases at the rate of $2 \text{inches/sec}$. How fast is the area increasing when the radius is $5$ inches ?

The displacement $s$ of a particle at a time $t$ is given by $s=2t^3-5t^2+4t-3$. Find the velocity and the displacement at time $t=2 \sec$.

The displacement $s$ of a particle at a time $t$ is given by $s=2t^3-5t^2+4t-3$. Find the time (in seconds) when the acceleration is $14\mathrm{ft}/{\sec }^2$. 

The approximate value of $\cos 31°$ is (take $1°=0.0174$ )

The $x-$coordinate changes on the curve $y=3x^5+15x-8$ at the rate of $\frac{1}{5}\mathrm{units}/\sec$, $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ are the points on the curve at which the $y-$coordinate changes at the rate of $6 \mathrm{units}/\sec$, then the slope of $AB=$