Q. Find the rate of change of the area of a circle with respect to its radius $r$ when
(i) $r=3$ cm
(ii) $r=4$ cm

1. $6\pi ,8\pi$
2. $4\pi ,10\pi$
3. $5\pi ,8\pi$
4. $2\pi ,8\pi$

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

Q. A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2x+1)$. Find the rate of change of its volume with respect to $x$.

Q. The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width $y$ is increasing at the rate of $4$ cm/minute. When $x=8$ cm and $y=6$ cm, find the rates of change of the area of rectangle(in $c{m}^2$/min).

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

Q. A stone is dropped into a quiet lake and waves move in circles at the speed of $5cm/s$. At the instant when the radius of the circular wave is $8cm$, how fast is the enclosed area increasing?

Q. A particle moves along the curve $6y=x^3+2$. Find the points on the curve at which the $y$-coordinate is changing $8$ times as fast as the $x$-coordinate.

Q. Find the rate of change of the area of a circle with respect to its radius $r$ when
(i) $r=3$ cm
(ii) $r=4$ cm

Q. A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15cm$.

Q. The total revenue in Rupees received from the sale of $x$ units of a product is given by $R\left(x\right)=13x^2+26x+15$.
Find the marginal revenue when $x=7$.

Q. The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width y is increasing at the rate of $4$ cm/minute. When $x=8cm$ and $y=6cm$ find the rates of change of (a) perimeter, and (b) the area of the rectangle.

Q. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10cm$.

Q. The radius of a circle is increasing at the rate of $0.7cm/s$. What is the rate of increase of its circumference?

Q. The volume of a cube is increasing at the rate of $8c{m}^3/s$. How fast is the surface area increasing when the length of an edge is $12cm$?

Q. The radius of an air bubble is increasing at the rate of $\frac{1}{2}cm/s$. At what rate is the volume of the bubble increasing when the radius is $1cm$?

Q. The total cost $C\left(x\right)$ in Rupees associated with the production of $x$ units of an item is given by $C\left(x\right)=0.007x^3-0.003x^2+15x+4000$.
Find the marginal revenue when $x=17.$

Q. The rate of change of the area of a circle with respect to its radius $r$ at $r=6cm$ is.

1. $10\pi$
2. $12\pi$
3. $8\pi$
4. $11\pi$

Q. A balloon which always remains spherical has a variable radius. The rate at which its volume is increasing is $m\pi$ $c{m}^2$ with the radius when the later is $10$ cm. Find $m$

Q. Sand is pouring from a pipe at the rate of $12c{m}^3/s$. 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 $4cm.$

Q. The radius of a circle is increasing uniformly at the rate of $3cm/s$. Find the rate at which the area of the circle is increasing when the radius is $10cm$.