ML AGGARWAL CLASS 12 Chapter 6 Applications of derivative Exercise 6.1 (CBSE)

  Exercise 6.1 

Very Short Answer Type Question (1-11)

Question 1

Find the rate of change of the area of a circle with respect to its radius when the radius is 6 cm.

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Question 2

If the radius of a circle is increasing at the rate of 0.7 cm/sec. at what rate is its circumference increasing ?

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Question 3

If the radius of a circle is increasing at the rate of 3 crn/sec. at what rate is its area increasing when its radius is 10 cm?

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Question 4

If the sides of a square are decreasing at the rate of 1.5 cm/sec, at what rate is its perimeter decreasing?

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Question 5

If the sides of an equilateral triangle are increasing at the rate of 2 cm/sec, at what rate is its area increasing when its side is 10 cm?

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Question 6

If an edge of a variable cube is increasing at the rate of 0.5 cm/sec, at what rate is its surface area increasing when its edge is 12 cm?

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Question 7

If an edge of a variable cube is increasing at the rate of 3 cm/sec, at what rate is its volume increasing when its edge is 10 cm?

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Question 8

A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with respect to radius when the radius is 10 cm. 

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Question 9

If the radius of a soap bubble is increasing at the rate of $\dfrac{1}{2}$ cm/sec ,at what rate is its volume increasing when the radius is 1 cm?

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Question 10

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Find the rate at which the depth of the wheat is increasing, take π = 3.14.

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Question 11

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.

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Question 12

Find the rate of change of area of a circle of radius r with regard to its radius. How fast is the area changing with respect to the radius when the radius is 5 cm?

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Question 13

Find the rate of change of the surface area of a sphere of radius r with respect to change in the radius when radius is 2 m.

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Question 14

Find the rate of change of the whole surface of a closed circular cylinder of radius r and height h with respect to change in radius.

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Question 15

A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and $L = 200(10 – t)^2$.

How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during first 5 seconds? 

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Question 16

(i) A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/sec. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? 

(ii) 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? 

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Question 17

The area of a circle of radius r increases at the rate of 5 $cm^2/sec$ ; find the rate at which the radius increases. Also find the value of this rate when the circumference is 10 cm.

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Question 18

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 rate of change of

(i) The perimeter

(ii) The area of the rectangle

Question 19

(i) The volume of a cube is increasing at the rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of the edge is 10 centimetres?

(ii) The volume of a cube is increasing at a constant rate. Prove that the increase in its surface area varies inversely as the length of an edge of the cube.

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Question 20

The volume of a spherical balloon is increasing at the rate of 20 $cm^3/sec$. Find the rate of change of its surface area when its radius is 8 cm.

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Question 21

The surface of a spherical balloon is increasing at the rate of 2 $cm2/sec$. Find the rate of change of its volume when its radius is 6 cm.

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Question 22

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

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Question 23

(i) Find the point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate.

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

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Question 24

Find an angle θ, $0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine.

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Question 25

The top of a ladder 6 metres long is resting against a vertical wall. Suddenly, the ladder begins to slide outwards. At the instant when the foot of the ladder is 4 metres from the wall, it is sliding away at the rate of 0.5 m/sec. How fast is the top sliding downwards at this moment? How far is the foot from the wall when the foot and the top are moving at the same rate?

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Question 26

A girl of height 1.6 m walks at the rate of 50 metres per minute away from a lamp which is 4 m above the ground. How fast is the girl’s shadow lengthening?

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Question 27

A kite is 120 m high and 130 m of string is out. If the kite is moving horizontally at the rate of 5.2 m/sec, find the rate at which the string is being paid out at that instant.

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Question 28

A circular cone, with semi-vertical angle 45°, is fixed with its axis vertical and its vertex downwards. Water is poured into the cone at the rate of 2 $cm^3$ per second. Find the rate at which the depth of the water is increasing when the depth is 4 cm.

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Question 29

Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

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Question 30

A conical vessel whose height is 4 metres and of base radius 2 metres is being filled with water at the rate of 0.75 cubic metres per minute. Find the rate at which the level of the water is rising when the depth of water is 1.5 metres. 31. 

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Question 31

Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the semi-vertical angle of the cone is $\dfrac{\pi}{6}$

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Question 32

$p(x) = 0.03 x^3 + 0.2 x^2 + 15 x + 100$ represents the air pollution in an industrial area due to smoke produced by x chimneys. Find the marginal value of air pollution when 3 chimney are increased. Which value does this question indicate?

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