2) The side of a square sheet is increasing at the rate of 3 cm per minute, when heated. At what rate is the area increasing when the side is 10 cm long ? 6 cm²/min
3) 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. 0.8 cm/sec
4) 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 ? 900cm³/s
5) A metal cube expands on heating. At the instant when each edge measures 3cm, the volume of the cube increases at the rate of 0.015 cm³/sec. At what rate is the length of the edge increasing at that instant. 0.00056 cm/sec
6) A hemisphere is constructed on a circular base. If the radius of the base is increasing at the rate of 0.5 cm/sec., Find the rate at which the volume of the hemisphere when its radius is 10cm. 314cm³/sec
7) The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference ? 1.4π cm/sec
8) 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 7cm. 11.2π cm²/sec
9) The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its volume when the radius is 5cm. 20π cm³/sec
10) A balloon which always remains spherical, is being inflated by pumping in 900 cm³ of a gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15cm. 1/π cm/sec
11) A spherical balloon is inflated so that its volume increases uniformly at the rate of 40 cm³/min. How fast is its surface area increasing when the radius is 8cm. Find approximately how much the radius will increase during next 1/2 minute. 10 cm²/min, 0.0249cm
12) 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 ? 2π cm³/sec
13) 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 10cm, how fast is the enclosed area increasing ? 80π cm²/sec.
14) A particle is moving in a vertical line whose equation of motion is s= 8+92t- 4.9t², when s is measured in metres and t is measured in seconds. Find the velocity and acceleration of the particle at t= 3. 62.6 m/sec, retardation= 9.8 m²/sec.
15) If y= 7x - x³ and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x= 2 ? 48
16) A particle moves along the curve y= x³. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-cordinate. (1,1),(-1,-1).
17) A particle moves along the curve y= x² + 2x. At what point/s on the curve are the x and y coordinates of the particle changing at the same rate? (-1/2, -1/3)
18) A particle moves along the curve 6y= x³ + 2. At what point/s on the curve the y coordinates is changing 8 times as fast as x-cordinate. (4, 11) and (-4,-31/3)
19) find an angle ¢, which increases twice as fast as its sine. π/3
20) A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter 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. 12π cm³/sec
21) The radius of a cylinder is increasing at the rate 2 cm/sec and its altitude is decreasing at the rate of 3 cm per second. find the rate of change of volume when radius is 3 cm and altitude 5 cm. 33π cm³/sec
22) 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 4cm and 8 cm respectively. 1/4 cm/sec
23) A kite is 120m high and 130m of string is out. If the kite is moving away horizontally at the rate of 52 metre per second, find the rate at which the string is being paid out. 20 m/sec
24) A particle moves along the curve y= 2x³/3 +1. Find the points on the curve at which the y-coordinate is changing twice as fast as the x-cordinate. (1,5/3),(-1,1/3)
25) find the point on the curve y²= 8x for which the abscissa and ordinate change at the same rate. (2,4)
26) The volume of a cube is increasing at the rate of 9 cm³/sec. How fast is the surface are increasing when the length of an edge is 10 cm ? 3.6 cm²/sec
27) 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=10cm and y = 6cm, find the rate of change of
A) the perimeter -2 cm/min
B) the area of rectangle. 2cm/m
28) The volume of a spherical balloon is increasing at the rate of 25 cm³/sec. Find the rate of change of its surface area at the instant when the radius 5 cm. 10 cm²/sec
29) 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 3m away from the base of light ? 0.5 m/sec
30) A ladder 13m 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/se . How fast is the angle ¢ between the ladder and the ground is changing when the foot of the ladder is 12m away from the wall. 0.3 radian/sec
31) A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres heigh. Find the rate at which the length of his shadow increases. 5/2 km/h
32) A man 160 cm tall, walks away from a source of light situated at the top of a pole 6m heigh, at the rate of 1.1m/sec. How fast is the length of his shadow increasing when he is 1 metre away from the pole ? 0.4 m/sec
33) A pole 13 metres long rests on a vertical wall. If the foot of the pole slips away from the wall with a constant velocity of 2.5 m/sec. Find at what rate its top is moving when the foot of the pole is at a distance of 12 metres from the wall. Find also the rate of which the slope of the pole changes at that instant. 6 m/sec, 0.5868 per second.
33) The top of a ladder 6 m long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwords. At the moment when the foot of the ladder is 4m 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 ? 1/√5 m/sec, 3√2 m
34) A man 2m high walks at a random speed of 6 km/h away from a lamp-post 6 metres high. Find rate at which the length of his shadow increases. 3 km/hr
35) A man is walking at the rate of 4.8 km/hr towards the foot of a tower 40 metres high. At what rate is he approaching the top of the when he is 30 metres away from the foot of the tower ? 2.88 km/hr
36) Water is the running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 m, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 metre above the base. 0.64 m/min
37) The surface of a spherical bubble is increasing at the rate of 2 cm²/s. When the radius of the bubble is 6cm, at what rate is the volume of the bubble increasing ? 6 cm³/sec
38) Sand is being poured onto a conical pile at the constant rate of 50 cm³/min 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. 1/2π cm/min.
39) An inverted cone has a depth of 10cms and a base radius 5 cms. Water is poured into it at the rate of 1.5 cm³ per minute. Find the rate at which the level of water in the cone is rising when the depth is 4 cms. 3/8π cm/min
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