Instantaneous Rate of Change

Q. The position of a particle moving along x- axis is given by x= 10t-2t^2. Then the time (t) at which it will momently come to rest is (a)0 (b)2.5s (c)5s (d)10s

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.

A spherical iron ball $10cm$ in radius is coated with a layer of ice of uniform thickness that metals at a rate of $50c{m}^3/min$. when the thickness of ice is $15cm$, then the rate of which the thickness of ice decreases, is

1. $\frac{1}{54\pi }cm/min$

2. $\frac{5}{6\pi }cm/min$

3. $\frac{1}{36\pi }cm/min$

4. $\frac{1}{18\pi }cm/min$

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 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

Q.

Rate of change is always measured with respect to time

1. True

2. False

Q.

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

1. $2x^2+3x^2+1$

2. $3x^2+2x^2-1$

3. $2x^2-3x+1$

4. $3x^2+2x+1$

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

Q. The angle of elevation of a jet plane from a point $A$ on the ground is ${60}^{\circ }.$ After a flight of $20$ seconds at the speed of $432\text{km/hour},$ the angle of elevation changes to ${30}^{\circ }.$ If the jet plane is flying at a constant height, then its height is

1. $1800\sqrt{3}$ m
2. $2400\sqrt{3}$ m
3. $1200\sqrt{3}$ m
4. $3600\sqrt{3}$ m

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.

A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the ballon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the ballon 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.

four hollw cubical boxes each having outer adges 1.0 m are joined to form a double bed in which top face is a squre.If the wood used is of thickness 5 cm and all the four boxes can be opened, then what is the capacity of the boxes of the double bed in cubic meteres?

Q. A box with square base and no top is to hold a volume of V. Find in terms of V the dimensions of the box that requires the least material for the 5 sides.Also find the ratio of height to the side of the base (this ratio will not involve V).