Rate of Change

Q. A closed cylinder with a given volume $V$ and minimum surface area has height equal to $k$ times the radius. Then value of $k$ is

Q.

For what values of x is the rate of increase of $x^3-5x^2+5x+8$ is twice rate of increase of x ?

1. $-3,-\frac{1}{3}$

2. $-3,\frac{1}{3}$

3. $3,-\frac{1}{3}$

4. $3,\frac{1}{3}$

Q.

A man of height 2m walks directly away from a lamp of height 5m, on a level road at 3 m/s. The rate at which the length of his shadow is increasing is

1. 4m/s

2. 1m/s

3. 2m/s

4. 3m/s

Q.

Suppose that water is emptied from a spherical tank of radius 10 cm. If the depth of the water in the tank is 4 cm and is decreasing at the rate of 2 cm/sec, them the radius of the top surface of water is decreasing at the rate of

1. 2

2. 1

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

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

Q.

A man is moving away from a tower 41.6m high at a rate of 2 m/s. If the eye level of the man is 1.6m above the ground, then the rate at which the angle of elevation of the top of the tower is changing when he is at a distance of 30 m from the foot of the tower is

1. None of these

2. $-\frac{4}{125}radian/sec$

3. $-\frac{2}{25}radian/sec$

4. $-\frac{1}{625}radian/sec$

Q. A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa is

1. $(\frac{9}{8},-\frac{9}{2})$
2. $(\frac{9}{8},\frac{9}{2})$
3. $(2,6)$
4. $(2,-6)$

Q. If a spherical balloon has a variable diameter $3x+\frac{9}{2},$ 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. Two people leave the same city to different destinations at the same time. Person $A$ leaves due east at a constant speed of $3\sqrt{2}$ kmph and person $B$ leaves due south east at a constant speed of $6$ kmph. How fast is the distance between them incresing $4$ hours later?

1. $12\sqrt{2}$
2. $6$
3. $3\sqrt{2}$
4. $24$

Q. The angle of intersection of the normals at the point $(-\frac{5}{\sqrt{2}},\frac{3}{\sqrt{2}})$ of the curves $x^2-y^2=8$ and $9x^2+25y^2=225$ is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $0$

Q. The range in meters of a projectile launched over a flat ground from the origin with positive velocity $V$ in $m/s$ at an angle $\theta$ given in radian is given by $R=\frac{V^2\sin \left(2\theta \right)}{g}$ where $g$ is a positive constant, assume $V=2m/s,g=10m/s^2$ and $\theta$ was measured to be $\frac{\pi }{12}$ radians. If there was a possible error in the measurement of $\theta$ of $\frac{1}{10\sqrt{3}}$ radians, estimate the corrosponding error in the computation of the range.

1. $0.04m$
2. $0.4m$
3. $0.08m$
4. $0.8m$