Rate of Change

Q.

The radius of a cylinder is increasing at the rate of $3m/s$ and its altitude is decreasing at the rate of $4m/s$. The rate of change of volume, when radius is $4m$and altitude is $6m$, is

1. $80\pi cum/s$

2. $144\pi cum/s$

3. $80cum/s$

4. $64cum/s$

Q.

The radius of a cylinder is increasing at the rate of $5cm/min$, so that its volume is constant. When its radius is $5cm$and height is $3cm$, then the rate of decreasing of its height is

1. $6cm/min$

2. $3cm/min$

3. $5cm/min$

4. $2cm/min$

Q.

A man of $2$ m height walks at a uniform speed of $6$ km/h, away from a lamp post of $6$ m height. The rate at which the length of his shadow increase is

1. $2$ km/h

2. $1$ km/h

3. $3$ km/h

4. $6$ km/h

Q.

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius$.$When the radius is $1cm$ the altitude is $6cm.$ When the radius is $6cm,$ then volume is increasing at the rate of $1c{m}^3/sec.$ When the radius is $36cm,$ the volume is increasing at a rate of $nc{m}^3/sec.$ The value of $n$ is equal to$:$

1. $12$

2. $22$

3. $30$

4. $33$

Q.

A train covers $300km$ in $4$hours . Find the distance covered by it in $9$hours.

Q.

If in a $△ABC,r_1=2,r_2=3,$ and $r_3=6$, then $a$ equals

1. $4$

2. $1$

3. $2$

4. $3$

Q.

A person who is $5$ feet tall casts a$6-foot$-long shadow. A nearby flagpole casts a $30-foot$-long shadow. What is the height of the flagpole?

1. $25$ ft

2. $29$ ft

3. $36$ ft

4. $40$ ft

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. 1

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

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

4. 2

Q. The radius of a right circular cylinder increases at the rate of $0.1\text{cm/min}$, and the height decreases at the rate of $0.2\text{cm/min}$. The rate of change of the volume of the cylinder, in ${\text{cm}}^3/\text{min}$, when the radius is $2\text{cm}$ and the height is $3\text{cm}$ is
(The negative sign(-) indicates that volume decreases)

1. $-\frac{2\pi }{5}$
2. $\frac{8\pi }{5}$
3. $-\frac{3\pi }{5}$
4. $\frac{2\pi }{5}$

Q. The triangle $ABC$ has angle $B={90}^{\circ }$. When it is rotated about $AB$, it gives a cone of volume $800\pi$. When it is rotated about $BC$, it gives a cone of volume $1920\pi$. The length of $AC$ is

Q.

Side of an equilateral triangle expands at the rate of 2 cm/s. The rate of increase of its area when each side is 10 cm, is

1. $10\sqrt{2}c{m}^2/sec$
   

2. $10\sqrt{3}c{m}^3/sec$
   

3. $10c{m}^2/sec$

4. $5c{m}^2/sec$

Q.

The legs of a right triangle have the proportion $8:15$, and the hypotenuse is $6.8cm$ long. Find the area of the triangle $?$

Q. A man of height $2$ metre walks at uniform speed of $3$ metre per second away from the lamp post of height $5$ metre. Then the rate at which the length of his shadow increases is $\text{m/sec}$

Q. A spherical iron ball of radius $10\text{cm}$ is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{cm}}^3\text{/min}.$ When the thickness of ice is $5\text{cm}$, then the rate at which the thickness $\left(\text{in}\text{cm/min}\right)$ of the ice decreases, is:

1. $\frac{1}{36\pi }$
2. $\frac{5}{6\pi }$
3. $\frac{1}{18\pi }$
4. $\frac{1}{9\pi }$

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. $0.8m$

Q. $f\left(x\right)=4+5x^2+6x^4$ has

1. only one minimum
2. neither maximum nor minimum
3. only one maximum
4. two local minimum

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. $0.8m$

Q. A spherical iron ball of radius $10\text{cm}$ is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{cm}}^3\text{/min}.$ When the thickness of ice is $5\text{cm}$, then the rate at which the thickness $\left(\text{in}\text{cm/min}\right)$ of the ice decreases, is:

1. $\frac{1}{36\pi }$
2. $\frac{5}{6\pi }$
3. $\frac{1}{18\pi }$
4. $\frac{1}{9\pi }$

Q. The radius of a right circular cylinder increases at the rate of $0.1$ cm/min, and the height decreases at the rate of $0.2$ cm/min. The rate of change of the volume of the cylinder, in ${\text{cm}}^3/\text{min},$ when the radius is $2$ cm and the height is $3$ cm, is

1. $-2\pi$
2. $\frac{8\pi }{5}$
3. $\frac{-3\pi }{5}$
4. $\frac{2\pi }{5}$

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. $24$