Instantaneous Rate of Change

Q.

If a spherical balloon has a variable diameter $3x+(9/2)$, then the rate of change of its volume with respect to $x$ is

1. $27\pi {(2x+3)}^2$

2. $\frac{27\pi }{16}{(2x+3)}^2$

3. $\frac{27\pi }{8}{(2x+3)}^2$

4. None of these

Q.

A spherical balloon is expanding. If the radius is increasing at the rate of $2cm/min$, then the rate at which the volume increases $(incm/min)$, when the radius is $5cm$, is

1. $10\mathrm{\pi }$

2. $100\mathrm{\pi }$

3. $200\mathrm{\pi }$

4. $50\mathrm{\pi }$

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.

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.

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. $8\pi c{m}^{2}sec$

2. $12\pi c{m}^{2}sec$

3. $160\pi c{m}^{2}sec$

4. $200\pi c{m}^{2}sec$

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. Suppose that water is being emptied from a spherical tank of radius $10$ m. At a particular instance of time if the radius of the water in the tank is $6$ m and is deceasing at the rate of $2$ m/s then, at what rate is the depth of the water in the tank decreasing?

1. $\frac{3}{2}$ m/s
2. $-\frac{3}{2}$ m/s
3. $\frac{3}{4}$ m/s
4. $-\frac{3}{4}$ m/s

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