Instantaneous Rate of Change

Q.

A hall is $15m$ long and $12m$ broad, the sum of the areas of the floor and the ceiling is equal to the sum of the areas of the four walls.

Find the volume of the hall?

Q. The volume of a cube is increasing at a rate of 7 cubic cm per second. The rate of change of its surface area when the length of an edge is 12 cm is

1. $\frac{7}{3}{\mathrm{cm}}^2/\sec .$
2. $\frac{3}{7}{\mathrm{cm}}^2/\sec .$
3. $\frac{7}{5}{\mathrm{cm}}^2/\sec .$
4. $\frac{5}{7}c{m}^2/sec.$

Q. Water is dripping out from a conical funnel of semi-vertical angle π/4 at the uniform rate of 2 square cms /sec in the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is 4 cm, then the rate of decrease of the slant height of water is

1. $\frac{\sqrt{2}}{4\pi }\mathrm{cm}/\sec .$
2. $\frac{\sqrt{2}}{2\pi }\mathrm{cm}/\sec .$
3. $\frac{3\sqrt{2}}{4\pi }\mathrm{cm}/\sec .$
4. $\frac{1}{4\sqrt{2}\pi }\mathrm{cm}/\sec .$

Q. The volume of a cube is increasing at a rate of $9$ cubic centimetres per second. How fast is the surface area increasing when the length of an edge is $10$ centimetres?

Q.

If u, v and w are functions of x, then show that $\frac{d}{dx}(u.v.w)=\frac{du}{dx}v.w+u.\frac{dv}{dx}w+u.v\frac{dw}{dx}$In two ways-first by repeated application of product rule, second by logarithmic differentiation.

Q.

The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm, is

1. 8π cm²/sec

2. 12π cm²/sec

3. 160π cm²/sec

4. 200π cm²/sec

Q. Air is being pumped into a spherical balloon at a constant rate such that its radius increases constantly with respect to time according to the equation $r\left(t\right)=0.5t^2+r_{\circ },$ (where $r$ is in $\text{cm},t$ is in $\text{minute}$ and $r_{\circ }$ is the initial radius of the balloon). Then the rate of change of its volume after $2\text{minute}$ is

(Assume that the initial radius of the balloon is $2\text{cm}$)

1. $32\pi {\text{cm}}^3\text{/min}$
2. $16\pi {\text{cm}}^3\text{/min}$
3. $128\pi {\text{cm}}^3\text{/min}$
4. $64\pi {\text{cm}}^3\text{/min}$

Q.

A photo is printed on an $11$-$inch$ by $13$-$inch$ piece of paper. The photo covers $80squareinches$ and has a uniform border. What is the width of the border?

Q.

A spherical balloon is pumped at the rate of $10inc{h}^3/min$, the rate of increase of its radius if its radius is 15 inch is

1. $\frac{1}{30\pi }inch/min$

2. $\frac{1}{60}inch/min$

3. $\frac{1}{90}inch/min$

4. $\frac{1}{120}inch/min$

Q.

The area of the base of a right circular cone is $314{\mathrm{cm}}^2$ and its height is $15\mathrm{cm}$. Find the volume of the cone.

Q.

$\mathrm{s}=\sqrt{({\mathrm{at}}^2+2\mathrm{bt}+\mathrm{c})}$ is the distance traveled by a particle traveling in a straight line at the time $\mathrm{t}$. The particle is accelerated

1. proportional to t

2. proportional to s

3. proportional to s^-3

4. none of these

Q.

A box with a square base and open top must have a volume of $32000{\mathrm{cm}}^3$.

How do you find the dimensions of the box that minimize the amount of material used?

Q.

If x and y are the sides of two squares such that $y=x-x^2$, then find the rate of change of the area of second square with respect to the area of first square.

Q.

A spherical iron ball of radius $10cm$, coated with a layer of ice of uniform thickness, melts at a rate of $100\pi cubiccm/min$. The rate at which the thickness layer decreases, when the thickness of ice is $5cm$, is

1. $1cm/min$

2. $2cm/min$

3. $\frac{1}{376}cm/min$

4. $5cm/min$

5. $3cm/min$

Q. Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is ${\cot }^{-1}\left(\sqrt{2}\right)$. [CBSE 2014]

Q.

What will be the number of unit squares on each side of a square graph paper if the total number of unit squares is $256$?

Q.

If a particle moving along a line follows the law $s=\sqrt{1+t}$ then the acceleration is proportional to

1. Square of the velocity

2. Cube of the displacement

3. Cube of the velocity

4. Square of the displacement

Q. The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area is

1. 2
2. infinite
3. 0
4. 1

Q. A particle moves along a straight line OX . At a time t(in seconds) the distance x=40+12t-t . How long would the particle travel before coming to rest?

Q.

Rate of change is always measured with respect to time

1. True

2. False

Q.

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.

Q.

The distance moved by the particle in time t is given by $x=t^3-12t^2+6t+8$. At the instant when its acceleration is zero, then the velocity is

1. 48

2. -48

3. 42

4. -42

Q.

The length, breadth, and height of a room are $5\mathrm{m},4\mathrm{m}\mathrm{and}3\mathrm{m}$respectively. Find the cost of whitewashing the walls of the room and ceiling at the rate of $\mathrm{Rs}.750\mathrm{per}{\mathrm{m}}^2$.

Q.

The volume of cube is increasing at the constant rate of 3 cm³/s. Find the rate of change of edge of the cube when its edge is 5 cm.

1. 

2. 

3. 

4. 

Q.

Two cylinder cans have bases of the same size. The diameter of each is $\mathbf{14}\mathbf{}\mathbf{cm}$. One of the cans is $\mathbf{10}\mathbf{}\mathbf{cm}$ high and the other is $\mathbf{20}\mathbf{}\mathbf{cm}$ high. Find the ratio of their volumes.

Q.

What is the rate of change of volume of a cube with respect to an edge when the diagonal of the cube is $6\sqrt{3}$ ?

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Q.

A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, the rate in meter per minute at which the radius of the balloon decreases $49$ minutes after the leakage began is

1. $\frac{9}{7}$

2. $\frac{7}{9}$

3. $\frac{2}{9}$

4. $\frac{9}{2}$

Q.

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Q.

A resort features a square pyramid with a water slide. The length of the water slide is $90\%$ of the height of the pyramid. The base of the pyramid has side lengths of $60feet$. The volume of the pyramid is $60,000cubicfeet$. What is the length of the water slide?

Q. The distance moved by a particle travelling in a straight line in t seconds is given by $s=45t+11t^2-t^3$. the time taken by the particle to come to rest is .

1. 9 seconds
2. 5/3 seconds
3. 3/5 seconds
4. 2 seconds