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Physics wallah Study Material

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[topic] Rate Measure

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In this topic, we will learn about change in any quantity w.r.t. other quantity.

#Q. Given, side length of a square changes at a rate of 2 cm/sec. Find the rate
of change in its area when side leg is 15 cm.

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4. Given, radius of a circle increases at a rate of 5 cm/sec. Find rate of increases in
(a) Rate of its area when radius is 20 cm.
(b) | Perimeter when radius is 12 cm.

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The length x of a rectangle is decreasing at the rate of 3 cm/min and the width
y is increasing at the rate of 2 cm/min. When x = 10 cm and y = 6cm, find the

rate of change of
() Area oO) P- 20x +y)

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6. A balloon which always remain spherical on inflation is being inflated by pumping in

900 cm? of gas per second. Find the rate at which the surface area of the balloon
increases when radius is 15 cm.

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7. (QUESTION) @
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A ladder 5 m long is lying against a wall. The bottom of the ladder is pulled along the
ground, away from the wall. At the rate of 2 cm/sec, how fast its height on the wall

decreases when foot of the ladder is 4 m away from the wall ?

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8. A particle moves along the curve 6y = x? + 4, find the points on the curve at
which the y-coordinate is changing 50 times as fast as the x-coordinate.

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9. (QUESTION) @
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The two equal sides of an isosceles triangle with fixed base ‘b’ are decreasing at

the rate of 3 cm/sec. How fast is the area decreasing when the two equal sides are
equal to the base?

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10. An inverted cone has a depth of 10 cm and a base of radius 5 cm. Water is poured into it

at a rate of 2 cm3/min. Find the rate at which the level of water in the cone is rising
when the depth is 4 em

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11. Water is dripping out from a conical funnel at a uniform rate 4 cm3/sec through tiny
hole at the vertex in the bottom. When the slant height of water is 3 cm, find the rate of
decrease of the slant height of water cone. Given that the vertical angle is 120°

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15. A man is walking at the rate of 6.5 km/hr towards the foot of a tower 120m high. At
what rate is he approaching the top of the tower, when he is 50 m away from the tower.

16. A man is moving away from a tower 41.6m high at the rate of 2 m/sec. Find rate at
which the angle of elevation of the top of tower is changing when he is at a distance of
30m from the foot of tower. Assume the level of the man is 1.6m from ground.

17. Let x be the length of the equal sides of an isosceles triangle and let O be the angle
between them. If x is increasing at the rate of 5 m/hr and 0 is increasing at the rate
of rad/hr then find the rate in m2/hr at which the area of triangle increases when

x= 12m.

18. (QUESTION) O
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A horse runs along a circle with a speed of 20 km/hr. A lantern is at the centre of the

circle. A fence is along the tangent to the circle at the point at which the horse starts,
find the speed with which the shadow of the horse moving along the fence at the

moment when it covers = th of the circle in km/hr.

19. A balloon, which always remains spherical, has a variable diameter 3 (ax+1). Find

the rate of change of its volume with respect to x.

20. A spherial balloon is filled wit 45007 cubic metres of helium gas. If a leak in the
balloon causes the gas to escape at the rate of 72x cubic metres per minute, then
the rate (in metres per minute) at which the radius of the balloon decreases 49
minutes after the leakage began is

EN 6/7
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21. If the surface area of a cube is increasing at a rate of 3.6 cm?/sec, retaining its
shape, then the rate of change of its volume (in cm?/sec.), when the length of a side
of the cube is 10 cm, is:

[M 2020]

22. If the volume of a spherical ball is increasing at the rate of 4x cc/sec, then the rate
of increase of its radius (in cm/sec), when the volume is 288 x cc,

UM 2014]

1/6

1/9

1/36

1/24

23. The total cost C(x) in Rupees associated with the production of x units of an item is
given by

C(x) = 0.007x3 - 0.003x2 + 15x + 4000.
Find the marginal cost when 17 units are produced.

24. Acylindrical 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

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25. Water is being poured at the rate of 1 cm? per minute into a cylindrical tub of base
radius 2.2 cm. The water level is rising at the rate of:

2x cm/min
0.07 cm/min

70 cm/min

none of these

26. @auestion) @
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Two cyclists start from the junction of two perpendicular roads, their velocities

being 3v meters/minute and 4v meters/minute. The rate at which the two cyclists
are separating is

[D.) none of these

27. The sides of the equilateral triangle are increasing at the rate of 2 cm/sec. The rate
at which the area increases, when the side is 10 cm

[c) 1043 sq. units/sec
CA 10/V3 sq. units/sec

28. ©

° DPP

DPP NO Question No

* Module
- Exercise Name

Module Page No Question No

29. O
(enopnama
=

1. Sub Top

Application of Derivatives

Questions

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30. 6

Solve the DPP and
check Solution

@ VIDYAPEETH

WORK, POWER AND ENERGY

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