Derivative as a Rate Measure

A man is moving away from a $40 \mathrm{m}$ high tower at a speed of $2 \mathrm{m}/\mathrm{s}$. Find the rate is which the angle of elevation of the top of the tower is changing when he is at a distance of $30$ metres from the foot of the tower. Assume that the eye level of the man is $1.6 \mathrm{m}$ from the ground.

A $13 \mathrm{m}$ long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2 \mathrm{m}/\mathrm{s}$. How fast is its height on the wall decreasing when the foot of the ladder is $5 \mathrm{m}$ away from the wall? 

Oil is leaking at the rate of $16 \mathrm{mL}/\mathrm{s}$ from a vertically kept cylindrical drum containing oil. If the radius of the drum is $7 \mathrm{cm}$ and its height is $60 \mathrm{cm}$, find the rate at which the level of the oil is changing when the oil level is $18 \mathrm{cm}$.

 The side of an equilateral triangle are increasing at the rate of $2 \mathrm{cm}/\sec$. Find the rate at which the area is increasing when the side is $10 \mathrm{cm}$.

An edge of a variable cube is increasing at the rate of $5 \mathrm{cm}/\mathrm{s}$. How fast is the volume of the cube increasing when the edge is $10 \mathrm{cm}$ long?

The radius of a balloon is increasing at the rate of $10 \mathrm{cm}/\sec$. At what rate is the surface area of the balloon increasing when the radius is $15 \mathrm{cm}$?

An angle $\frac{\mathrm{\pi }}{k}$ which increases twice as fast as its sine. Find the value of $k$.

Water is dripping through a tiny hole at the vertex in the bottom of a conical funnel at a uniform rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. When the slant height of the water is $3 \mathrm{cm}$, the rate of decrease of the slant height of the water in $\mathrm{cm}/\mathrm{s}$, given that the vertical angle of the funnel is $120°$ is $\frac{a}{596} \mathrm{cm}/\mathrm{s}$. Find $a$. (Take $\mathrm{\pi }=\frac{22}{7}$)

Sand is pouring from a pipe at the rate of $18 {\mathrm{cm}}^3/\mathrm{s}$. The falling sand forms a cone on the ground in such a way that the height of the cone is one-sixth of the radius of the base. Height of the sand cone increasing when its height is $3 \mathrm{cm}$ is $\frac{a}{b} \mathrm{cm}/\sec$, where $a \& b$ are smallest positive integers. Find $a+b$. (Take $\mathrm{\pi }=\frac{22}{7}$)

An inverted cone has a depth of $40 \mathrm{cm}$ and a base of radius $5 \mathrm{cm}$. Water is poured into it at a rate of $1.5$ cubic centimetres per minute. The rate at which the level of water in the cone is rising when the depth is $4 \mathrm{cm}$ in $\mathrm{cm}/\sec$ is $\frac{a}{b} \mathrm{cm}/\sec$, where $a$ and$b$ are smallest positive integers . Find$a+b$.  (Take $\mathrm{\pi }=\frac{22}{7}$)

A $2 \mathrm{m}$ tall man walks at a uniform speed of $5 \mathrm{km}/\mathrm{h}$ away from a$6 \mathrm{m}$ high lamp post. Find the rate at which the length of his shadow increases in $\mathrm{km}/\mathrm{h}$.

A stone is dropped into a quiet lake and waves move in circles at a speed of $3.5 \mathrm{cm}/\mathrm{s}$. At the instant when the radius of the circular wave is $7.5 \mathrm{cm},$ if the enclosed area is increasing at the rate of $k {\mathrm{cm}}^2/\mathrm{s}$, then what is the value of $k$? (Take $\mathrm{\pi } = \frac{22}{7}$)

The radius of a circle is increasing at the rate of $0.7 \mathrm{cm}/\mathrm{s}$. What is the rate of increase of its circumference in $\mathrm{cm}/\mathrm{s}$? (Take $\mathrm{\pi }=\frac{22}{7}$)

The side of a square is increasing at the rate of $0.2 \mathrm{cm}/\mathrm{s}$. If the rate of increase of the perimeter of the square is $k \mathrm{cm}/\mathrm{s}$, then the value of $k$ is

The bottom of a rectangular swimming tank is $25 \mathrm{m}$ by $40 \mathrm{m}$. Water is pumped into the tank at the rate of $500$ cubic metres per minute. Find the rate at which the level of water in the tank is rising in $\mathrm{m}/\min$.

A balloon which always remains spherical is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate (in $\mathrm{cm}/\mathrm{s}$) at which the radius of the balloon is increasing when the radius is $15 \mathrm{cm}$. Use $\pi =3.14$        (Round off final answer to two decimal places and enter the value excluding units)

The volume of a spherical balloon is increasing at the rate of$25$cubic centimetres per second. Find the rate of change of its surface in ${\mathrm{cm}}^2/\mathrm{s}$ at the instant when its radius is $5 \mathrm{cm}$.

The radius of an air bubble is increasing at the rate of $0.5$ centimetre per second. At what rate is the volume of the bubble increasing in ${\mathrm{cm}}^3/\mathrm{s}$ when the radius is $1$ centimetre? (Take $\mathrm{\pi }=3.14$)

The radius of a spherical soap bubble is increasing at the rate of $0.2 \mathrm{cm}/\mathrm{s}$. If the rate of increase of its surface area when the radius is $7 \mathrm{cm}$ is $k {\mathrm{cm}}^2/\mathrm{s}$, then find the value of $k$. (Take $\mathrm{\pi }=\frac{22}{7}$)

{Write the answer in decimal form}

The side of a square sheet of a metal is increasing at a rate of $3$ centimetres per minute. At what rate is the area increasing in ${\mathrm{cm}}^2/\mathrm{s}$ when the side is $10 \mathrm{cm}$ long?