# RD SHARMA Solutions for Class 12-science Maths Chapter 13 - Derivative as a Rate Measurer

Page / Exercise

## Chapter 13 - Derivative as a Rate Measurer Exercise Ex. 13.1

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

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

Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10

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) received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x=5, and write which value does the question indicate.

Solution 10

## Chapter 13 - Derivative as a Rate Measurer Exercise Ex. 13.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

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.

Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8

Question 9
Solution 9
Question 10
Solution 10

Question 11
Solution 11

Question 12
Solution 12

Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16(i)

Find an angle θ, which increases twice as fast as its cosine.

Solution 16(i)

Question 16(ii)

Find the angle θ

Whose rate of increase is twice the rate of decrease of its consine.

Solution 16(ii)

Question 17
Solution 17

Question 18
Solution 18

Question 19
Solution 19

Question 20
Solution 20

Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25
Solution 25

Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31

## Chapter 13 - Derivative as a Rate Measurer Exercise MCQ

Question 1

Solution 1

Correct option:(b)

Question 2

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

Solution 2

Correct option: (b)

Question 3

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. 8 π cm2/sec

b. 12π cm2/sec

c. 160 πcm2/sec

d. 200 π cm2/sec

Solution 3

Correct option: (c)

Question 4

A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?

1. 1 mm/sec
2. 0.001 cm/sec
3. 2 mm/sec
4. 0.002 cm/sec
Solution 4

Correct option:(d)

Question 5

A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m3/minute. The rate at which the surface of the oil is rising, is

1. 1 m/minute
2. 2 m/minute
3. 5 m/minute
4. 1.25 m/minute
Solution 5

Correct option: (a)

Question 6

The distance moved by the particle in time t is given by x = t3 - 12t2 + 6t + 8. At the instant when its acceleration is zero, the velocity is

1. 42
2. -42
3. 48
4. -48
Solution 6

Correct option: (b)

Question 7

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

1. 30 cm/sec
2. 10 cm/sec
3. 160 cm/sec
Solution 7

Correct option: (b)

Question 8

For what value of x is the rate of increase of x3 - 5x2 + 5x + 8 is twice the rate increase of x?

1. -3, -1/3
2. -3, 1/3
3. 3, -1/3
4. 3, 1/3
Solution 8

Correct option: (d)

Question 9

The coordinate of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are

1. (3, 16/3)
2. (-3, 16/3)
3. (3, -16/3)
4. (3, -3)
Solution 9

Correct option: (a)

Question 10

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 = 7cm and altitude 24 cm is

a. 54π cm2/min

b. 7π cm2/min

c. 27π cm2/min

d. none of these

Solution 10

Correct option: (a)

Question 11

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

1. 12π cm3/sec
2. 180π cm3/sec
3. 225π cm3/sec
4. 3π cm3/sec
Solution 11

Correct option: (b)

Question 12

The volume of a sphere is increasing at 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is

Solution 12

Correct option: (b)

Question 13

The distance moved by a particle travelling in a straight line in a straight line in t seconds is given by s = 45t + 11t2 - t3. The time taken by the particle to come to rest is

1. 9 sec
2. 5/3 sec
3. 3/5 sec
4. 2 sec
Solution 13

Correct option: (a)

Question 14

The volume of a sphere is increasing at the rate of 4π cm3/sec. The rate of increase of the radius when the volume is 288π  cm3, is

a. 1/4

b. 1/12

c. 1/36

d. 1/9

Solution 14

Correct option: (c)

Question 15

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

Solution 15

Correct option: (d)

Question 16

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

1. 2/Π unit
2. 1/Π unit
3. Π/2 units
4. Π units
Solution 16

Correct option: (b)

Question 17

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

Solution 17

Correct option: (a)

Question 18

If s = t3 - 4t2 + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, is

1. 16/9 units/sec
2. -32/3 units/sec
3. 4/3 units/sec
4. -16/3 units/sec
Solution 18

Correct option: (d)

Question 19

The equation of motion of a particle is s = 2t2 + sin 2t, where s is in metres and t is in seconds. The velocity of the particle when its acceleration is 2m/sec2, is

Solution 19

Correct option: (b)

Question 20

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 cm2/sec

b. 2.4 π cm2/sec

c. 0.24 π cm2/sec

d. 0.024 π cm2/sec

Solution 20

Correct option: (c)

Question 21

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

1. π cm2/sec
2. 2π  cm2/sec
3. π2 cm2/sec
4. 2π2 cm2/sec2
Solution 21

Correct option: (c)

Question 22

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

1. 1.6 km/hr
2. 6.3 km/hr
3. 5 km/hr
4. 3.2 km/hr
Solution 22

Correct option: (d)

Question 23

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

1. 15 ft/sec
2. 9 ft/sec
3. 6 ft/sec
4. None of these
Solution 23

Correct option: (c)

Question 24

In a sphere the rate of change of volume is

1. π time the rate of change of radius
2. surface area times the rate of change of diameter
3. surface area times the rate of change of radius
4. none of these
Solution 24

Correct option: (c)

Question 25

In a sphere the rate of change of surface area is

1. 8π times the rate of change of diameter
2. 2π times the rate of change of diameter
3. 2π times the rate of change of radius
4. 8π times the rate of change of radius
Solution 25

Correct option: (d)

Question 26

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

1. 1m/hr
2. 0.1m/hr
3. 1.1 m/hr
4. 0.5 m/hr

Solution 26

Correct option: (a)

## Chapter 13 - Derivative as a Rate Measurer Exercise Ex. 13VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

The amount of pollution content added in air in a city due to x diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions.

Solution 9

Question 10

A ladder, 5 meter 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.

Solution 10

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