#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 13: Derivative as a Rate Measurer

#### Chapter 13: Derivative as a Rate Measurer solutions [Page 4]

Find the rate of change of the total surface area of a cylinder of radius *r* and height h, when the radius varies ?

Find the rate of change of the volume of a sphere with respect to its diameter ?

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?

Find the rate of change of the volume of a cone with respect to the radius of its base ?

Find the rate of change of the area of a circle with respect to its radius *r* when *r* = 5 cm ?

Find the rate of change of the volume of a ball with respect to its radius *r*. How fast is the volume changing with respect to the radius when the radius is 2 cm?

The total cost *C* (*x*) associated with the production of *x* units of an item is given by *C* (*x*) = 0.007*x*^{3} − 0.003*x*^{2} + 15*x* + 4000. Find the marginal cost when 17 units are produced ?

The total revenue received from the sale of *x* units of a product is given by *R* (*x*) = 13*x*^{2} + 26*x* + 15. Find the marginal revenue when *x* = 7 ?

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of *x* units of a product is given by *R*(*x*) = 3*x*^{2} + 36*x* + 5, find the marginal revenue, when *x* = 5, and write which value does the question indicate ?

#### Chapter 13: Derivative as a Rate Measurer solutions [Pages 19 - 21]

The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?

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

The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.

The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?

The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.

A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?

A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

A man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole?

A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?

A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle θ between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall.

A particle moves along the curve *y* = *x*^{2} + 2*x*. At what point(s) on the curve are the *x* and *y*coordinates of the particle changing at the same rate?

If *y* = 7*x* − *x*^{3} and *x* increases at the rate of 4 units per second, how fast is the slope of the curve changing when *x* = 2?

A particle moves along the curve *y* = *x*^{3}. Find the points on the curve at which the *y*-coordinate changes three times more rapidly than the *x*-coordinate.

Find an angle *θ *which increases twice as fast as its cosine ?

Find an angle *θ *whose rate of increase twice is twice the rate of decrease of its cosine ?

The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?

How far is the foot from the wall when it and the top are moving at the same rate?

A balloon in the form of a right circular cone surmounted by a hemisphere, having a diametre equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height *h*, when *h* = 9 cm.

Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.

A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?

The surface area of a spherical bubble is increasing at the rate of 2 cm^{2}/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?

The radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm.

The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.

Sand is being poured onto a conical pile at the constant rate of 50 cm^{3}/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep ?

A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out.

A particle moves along the curve *y* = (2/3)*x*^{3} + 1. Find the points on the curve at which the *y*-coordinate is changing twice as fast as the *x*-coordinate ?

Find the point on the curve y^{2} = 8*x* for which the abscissa and ordinate change at the same rate ?

The volume of a cube is increasing at the rate of 9 cm^{3}/sec. How fast is the surface area increasing when the length of an edge is 10 cm?

The volume of a spherical balloon is increasing at the rate of 25 cm^{3}/sec. Find the rate of change of its surface area at the instant when radius is 5 cm ?

The length *x* of a rectangle is decreasing at the rate of 5 cm/minute and the width *y* is increasing at the rate of 4 cm/minute. When *x* = 8 cm and *y* = 6 cm, find the rates of change of the perimeter ?

The length *x* of a rectangle is decreasing at the rate of 5 cm/minute and the width *y* is increasing at the rate of 4 cm/minute. When *x* = 8 cm and *y* = 6 cm, find the rates of change of the area of the rectangle ?

A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.

#### Chapter 13: Derivative as a Rate Measurer solutions [Page 24]

If a particle moves in a straight line such that the distance travelled in time *t* is given by *s* = *t*^{3} − 6*t*^{2}+ 9*t* + 8. Find the initial velocity of the particle ?

The volume of a sphere is increasing at 3 cubic centimeter per second. Find the rate of increase of the radius, when the radius is 2 cms ?

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. How far is the area increasing when the side is 10 cms?

The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?

The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?

The side of an equilateral triangle is increasing at the rate of \[\frac{1}{3}\] cm/sec. Find the rate of increase of its perimeter ?

Find the surface area of a sphere when its volume is changing at the same rate as its radius ?

If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?

The amount of pollution content added in air in a city due to *x* diesel vehicles is given by *P*(*x*) = 0.005*x*^{3} + 0.02*x*^{2} + 30*x*. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions ?

A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall ?

#### Chapter 13: Derivative as a Rate Measurer solutions [Pages 24 - 26]

If \[V = \frac{4}{3}\pi r^3\] , at what rate in cubic units is *V* increasing when *r* = 10 and \[\frac{dr}{dt} = 0 . 01\] ?

(a) π

(b) 4π

(c) 40π

(d) 4π/3

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

(a) \[10\sqrt{2} \ {cm}^2 /\sec\]

(b) \[10\sqrt{3} {cm}^2 /\sec\]

(c) 10 cm^{2}/sec

(d) 5 cm^{2}/sec

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

(a) 8π cm^{2}/sec

(b) 12π cm^{2}/sec

(c) 160π cm^{2}/sec

(d) 200 cm^{2}/sec

A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm^{3}/sec. At what rate is the radius increasing when its circular base area is 1 m^{2}?

(a) 1 mm/sec

(b) 0.001 cm/sec

(c) 2 mm/sec

(d) 0.002 cm/sec

A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m^{3}/minute. The rate at which the surface of the oil is rising, is

(a) 1 m/minute

(b) 2 m/minute

(c) 5 m/minute

(d) 1.25 m/minute

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

(a) 42

(b) −42

(c) 48

(d) −48

The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of

(a) 30 cm/sec

(b) \[\frac{160}{3} cm/\sec\]

(c) 10 cm/sec

(d) 160 cm/sec

For what values of *x* is the rate of increase of x^{3}^{ }− 5x^{2} + 5x + 8 is twice the rate of increase of x ?

(a) \[- 3, - \frac{1}{3}\]

(b) \[- 3, \frac{1}{3}\]

(c) \[3, - \frac{1}{3}\]

(d) \[3, \frac{1}{3}\]

The coordinates of the point on the ellipse 16*x*^{2} + 9*y*^{2} = 400 where the ordinate decreases at the same rate at which the abscissa increases, are

(a) (3, 16/3)

(b) (−3, 16/3)

(c) (3, −16/3)

(d) (3, −3)

The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius = 7 cm and altitude 24 cm is

(a) 54π cm^{2}/min

(b) 7π cm^{2}/min

(c) 27 cm^{2}/min

(d) none of these

The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is

(a) 12π cm^{3}/sec

(b) 180π cm^{3}/sec

(c) 225π cm^{3}/sec

(d) 3π cm^{3}/sec

The volume of a sphere is increasing at 3 cm^{3}/sec. The rate at which the radius increases when radius is 2 cm, is

(a) \[\frac{3}{32\pi}cm/\sec\]

(b) \[\frac{3}{16\pi}cm/\sec\]

(c) \[\frac{3}{48\pi}cm/\sec\]

(d) \[\frac{1}{24\pi}cm/\sec\]

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

(a) 9 sec

(b) 5/3 sec

(c) 3/5 sec

(d) 2 sec

The volume of a sphere is increasing at the rate of 4π cm^{3}/sec. The rate of increase of the radius when the volume is 288 π cm^{3}, is

(a) 1/4

(b) 1/12

(c) 1/36

(d) 1/9

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

(a) 1 unit

(b) \[\sqrt{2\pi} \text { units }\]

(c) \[\frac{1}{\sqrt{2\pi}} \text { unit }\]

(d) \[\frac{1}{2\sqrt{\pi}} \text { unit}\]

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to

(a) \[\frac{2}{\pi} \text { unit }\]

(b) \[\frac{1}{\pi} \text { unit }\]

(c) \[\frac{\pi}{2} \text { units }\]

(d) π units

Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is

(a) \[8\sqrt{3} \ {cm}^2 /hr\]

(b) \[4\sqrt{3} \ {cm}^2 /hr\]

(c) \[\frac{\sqrt{3}}{8} \ {cm}^2 /hr\]

(d) none of these

If *s* = *t*^{3} − 4*t*^{2} + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, is

(a) \[\frac{16}{9} \text { unit }/\sec\]

(b) \[- \frac{32}{3} \text { unit }/\sec\]

(c) \[\frac{4}{3} \text { unit }/\sec\]

(d) \[- \frac{16}{3} \text { unit }/\sec\]

The equation of motion of a particle is *s* = 2*t*^{2} + sin 2*t*, where *s* is in metres and *t *is in seconds. The velocity of the particle when its acceleration is 2 m/sec^{2}, is

(a) \[\pi + \sqrt{3} m\ /\sec\]

(b) \[\frac{\pi}{3} + \sqrt{3} m/\sec\]

(c) \[\frac{2\pi}{3} + \sqrt{3} m/\sec\]

(d) \[\frac{\pi}{3} + \frac{1}{\sqrt{3}} m/\sec\]

The radius of a circular plate is increasing at the rate of 0.01 cm/sec. The rate of increase of its area when the radius is 12 cm, is

(a) 144 π cm^{2}/sec

(b) 2.4 π cm^{2}/sec

(c) 0.24 π cm^{2}/sec

(d) 0.024 π cm^{2}/sec

The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is

(a) π cm^{2}/sec

(b) 2π cm^{2}/sec

(c) π^{2} cm^{2}/sec

(d) 2π^{2} cm^{2}/sec^{2}

A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is

(a) 1.6 km/hr

(b) 6.3 km/hr

(c) 5 km/hr

(d) 3.2 km/hr

A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp fixed at 15 ft height. The length of his shadow is increasing at the rate of

(a) 15 ft/sec

(b) 9 ft/sec

(c) 6 ft/sec

(d) none of these

In a sphere the rate of change of volume is

(a) π times the rate of change of radius

(b) surface area times the rate of change of diameter

(c) surface area times the rate of change of radius

(d) none of these

In a sphere the rate of change of surface area is

(a) 8π times the rate of change of diameter

(b) 2π times the rate of change of diameter

(c) 2π times the rate of change of radius

(d) 8π times the rate of change of radius

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

(a) 1 m/hr

(b) 0.1 m/hr

(c) 1.1 m/hr

(d) 0.5 m/hr

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 13 - Derivative as a Rate Measurer

RD Sharma solutions for Class 12 Maths chapter 13 (Derivative as a Rate Measurer) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 13 Derivative as a Rate Measurer are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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