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

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

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

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) = 3x^{2}
+ 36x + 5, find the marginal revenue, when x=5, and write which value does
the question indicate.

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

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.

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

Find the angle θ

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

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

Correct option:(b)

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

Correct option: (b)

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 cm^{2}/sec

b. 12π cm^{2}/sec

c. 160 πcm^{2}/sec

d. 200 π cm^{2}/sec

Correct option: (c)

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}?

- 1 mm/sec
- 0.001 cm/sec
- 2 mm/sec
- 0.002 cm/sec

Correct option:(d)

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

- 1 m/minute
- 2 m/minute
- 5 m/minute
- 1.25 m/minute

Correct option: (a)

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

- 42
- -42
- 48
- -48

Correct option: (b)

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

- 30 cm/sec
- 10 cm/sec
- 160 cm/sec

Correct option: (b)

For what value of x is the rate of increase of x^{3} - 5x^{2 }+ 5x + 8 is twice the rate increase of x?

- -3, -1/3
- -3, 1/3
- 3, -1/3
- 3, 1/3

Correct option: (d)

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

- (3, 16/3)
- (-3, 16/3)
- (3, -16/3)
- (3, -3)

Correct option: (a)

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 cm^{2}/min

b. 7π cm^{2}/min

c. 27π cm^{2}/min

d. none of these

Correct option: (a)

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

- 12π cm
^{3}/sec - 180π cm
^{3}/sec - 225π cm
^{3}/sec - 3π cm
^{3}/sec

Correct option: (b)

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

Correct option: (b)

The distance moved by a particle travelling in a straight line 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

- 9 sec
- 5/3 sec
- 3/5 sec
- 2 sec

Correct option: (a)

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 288π cm^{3}, is

a. 1/4

b. 1/12

c. 1/36

d. 1/9

Correct option: (c)

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

Correct option: (d)

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

- 2/Π unit
- 1/Π unit
- Π/2 units
- Π units

Correct option: (b)

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

Correct option: (a)

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

- 16/9 units/sec
- -32/3 units/sec
- 4/3 units/sec
- -16/3 units/sec

Correct option: (d)

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

Correct option: (b)

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

Correct option: (c)

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

- π cm
^{2}/sec - 2π cm
^{2}/sec - π
^{2}cm^{2}/sec - 2π
^{2}cm^{2}/sec^{2}

Correct option: (c)

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

- 1.6 km/hr
- 6.3 km/hr
- 5 km/hr
- 3.2 km/hr

Correct option: (d)

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

- 15 ft/sec
- 9 ft/sec
- 6 ft/sec
- None of these

Correct option: (c)

In a sphere the rate of change of volume is

- π time the rate of change of radius
- surface area times the rate of change of diameter
- surface area times the rate of change of radius
- none of these

Correct option: (c)

In a sphere the rate of change of surface area is

- 8π times the rate of change of diameter
- 2π times the rate of change of diameter
- 2π times the rate of change of radius
- 8π times the rate of change of radius

Correct option: (d)

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

- 1m/hr
- 0.1m/hr
- 1.1 m/hr
- 0.5 m/hr

Correct option: (a)

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

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 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.

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