1. The sides of a square are increasing at the rate of 0.2 m per second. Find at what rate (i) the area (ii) the length of the diagonal is increasing when the side is 8 m. [ans: 3.2; 0.28]

2. The sides of a cube are increasing at the rate of 0.4 cm per second. Find at what rate (i) the volume (ii) the surface area is increasing when the sides are each 10 cm. [ans: 120; 48].

3. A point moves along the curve . The abscissa of the point changes at the rate of 0.5 unit per second. At what rate is the ordinate changing when x = 4. [ans: 64].

4. The sides of an equilateral triangle are increasing at the rate of 1/6 cm per second. At what rate is the area increasring at the time when the sides are 12 cm long. (Area of an equilateral triangle is where x is the length of the side). [ans: ]

5. In a rectangle the length is always twice the width. If the width increases at the rate of 0.5 cm per minute, find the rate of change of (i) the length (ii) the area (iii) the perimeter if the width is 8 cm. [ans: 1; 16; 3]

6. The radius of a circular metal plate is increasing at the rate of 0.01 cm per sec. when being subjected to heat. AT what rate is the area increasing when the radius is 5 cm. [ans: 0.314]

7. A spherical balloon is expanding so that the radius increases at the rate of 0.5 cm per sec., at what rate is (i) the surface (ii) the volume increasing when the radius is 10.5 cm. [ans: 132; 693]

8. Find the point on the curve where the abscissa and the ordinate of the point are increasing at the same rate. [ans: (3.5; 15.75]

9. A cylindrical solid is being turned on a lathe so that the radius is being reduced by 0.4 cm per min., the height remaining unchanged at 10 cm. Find at what rate (i) the volume (ii) the surface area is decreasing when the radius is 14 cm. [ans: 352; 152p/5]

10. A vessel is in the form of an inverted cone with a vertical angle of 90o. If the depth of the water in the vessel is x cm, find the volume of water it contains in terms of x. If water is poured in at the rate of 38.5 cu.cm per min. , find at what rate the depth is increasing when it is 3.5 cm. [ans: V=px3/3]

11. A hemispherical bowl of radius 8 cm. has water pouring into it at the rate of 40 c.c per min. At what rate is the level of the water rising when the depth of the water in the bowl is 5 cm ? (The volume of a hemisphereical segment is , where r is the radius of the sphere, h is the altitude of the segment). [ans: 8/11p]

12. A parabolic shaped vessel is such that when it contains liquid to a depth of y cm, the volume of the liquid is cubic cm. Liquid is poured into the vessel at the rate 40 cubic cm per min., at what rate is the level of the liquid rising when the depth of the liquid is 6 cm. [ans:5/81]

13. A a given instant the radii of 2 concentric sphares are 6 cm and 7 cm. The radius of the smaller sphere is decreasing at the rate of 1 mm per min, while the radius of the larger sphere is increasing at the rate of 1.5 mm per min. Find the rate at which the volume between the sphere is increasing. [ans: 219p/5]

14. A spherical balloon of radius R and volume V has its radius increasing at the rate of 1% per sec. (i) At what percentage rate is the volume increasing; (ii) At what percentage rate is the surface area increasing ? [ans:3%; 2%]

15. The radius of the base of a cone is increasing at the rate of 2% per sec. At what percentage rate is the height diminishing if the volume is kepth constant ? [ans:4%]

16. A horizontal trough 10 m long, has a cross section in the shape of a right angled isosceles triangle. If water is poured in at the rate of 8 cu m per min., at what rate is the water level rising when the depth of the water is 2 m ? [ans: 0.2 m]

17. The volume of liquid in a vessel is given gy the formula where y is the depth of the liquid in cm. Water is being poured in at the rate of 64 cu cm per sec. Find the rate of change of the water level when y = 8 cm. [ans: 1/44]

18. The pressure p kilopascal (KPa) on a mass of gas of volume v cubic cm is given by the formula pv = 1500. If the volume increases at the rate of 60 cu.cm./min Find the rate at which the pressure is decreasing when the volume is 30 cubic cm. [ans:100 kPa]

19 (i) If , find dy/dt when x = 3 given that dx/dt = 7.

(ii) If and dx/dt is constant and equal to -3, find dy/dt when x = 2.

(iii) If and dx/dt = 3 find dy/dt when x = -8.

20 (i) If , find dx/dt when x = 2 given that dy/dt = 24.

(ii) If y = 6/x and dy/dt is constant and equal to -9 find dx/dt when y = 2.

(iii) If and dy/dt = -36 find dx/dt when x = 1.

21. The sides of a square are increasing at the rate of 0.5 cm/s. Find at what rate (a) the area (b) the length of a diagonal is increasing when the side is 8 cm.

22. The sides of a cube are increasing at the rate of 2 cm/s. Find at what rate (a) the volume (b) the surface area (c) the length of a diagonal is increasing when the sides are each 10 cm.

23. Heat is applied toi a circular plate so that it expands uniformly. When the radius of the plate is exactly 25 cm, it is increasing at the rate of 4 /s. Find the rate of increase of (a) the area (b) the circumference at this instant.

24. A spherical balloon is expanding so that the radius increases at the rate of 0.5 cm/s. At what rate is (a) the surface area (b) the volume increasing when the radius is 10 cm ?

25. If the area of a square plate is increasing at the constant rate of 72 /min, find the rate the sides are increasing when (a) each side is 4.5 cm (b) the area of the plate is 36 .

26 (i) Metal is being removed from a circular metal disc at the rate of 5 /s by a machine (the disc remaining circular). Find the rate at which the radius is decreasing at the instant when the radius is 3 cm.

(ii) The temperature of a cube is falling so that the volume decreases at a constant rate of 0.06 /s. Find at what rate the edge is contracting when the edge is 10 cm.

27 (i) Assuming that a soap bubble retains its spherical shape as it expands, find how fast the radius is increasing when its radius is 5 cm, if air is blown in it at the rate of 4 /s.

(ii) A point moves along the curve , so that the abscissa of the point increases at the constant rate of 0.5 units per second. Find at what rate of ordinate is changing when (a) x = 4 (b) y = -72.

28. A particle P moves on the curve .

(i) Find the velocity of P parallel to the axis Ox (the rate of change of its x-coordinate) at the instant when the x-coordinate is -2 and the velocity of P parallel to the axis Oy is 20 (this is the rate of increase of the y-coordinate).

(ii) Determine the coordinates of the point on the curve where the absissa and the ordinate of P are increasing at the same rate.

29. A parabolic shaped vessel is such that when it contains liquid to a depth of y cm, the volume of liquid is . Liquid is poured into the vessel at the rate of 40 /min. At what rate is the level of the liquid rising when the depth of the liquid is 6 cm ?

30. Prove that the area A cm2 of an equilateral triangle of side x cm is given by

A = .(hints: use sine/cosine/area of triangle rules).

The sides of an equilateral triangle are increasing at the rate of 1/6 cm/s. At what rate is the area increasing at the instant when the sides are 12 cm ?

31. In a rectangle, the length is always twice the breadth. If the breadth increases at the constant rate of 0.5 cm/min (the figure remaining a rectangle) find the area of change of (i) the length (ii) the area (iii) the perimeter at the instant when the breadth is 8 cm.

32 (i) A vertical cylinder of radius 9 cm has water poured into it at the rate of 2 /s. Find the rate at which the depth of the water increases.

(ii) The height of a cone is 15 cm and remains constant, whilst the radius of the base is increasing at the rate of 6 cm/min. At what rate is the volume of the cone increasing at the instant when the radius is 20 cm?

33. Gas is being pumped into a spherical balloon at the rate of 75 /min . When the radius is 25 m, find the rate at which the (i) radius (ii) surface area is increasing.

34. The surface area of a cube is increasing at the constant rate of 8 /s. At the instant when the edge is 6 cm, find the rate of increase of the (i) edge (ii) volume.

35. (i) The volume of a cube is increasing at the rate of 0.001 /s. Find how fast the surface area is changing when the side is 5 cm.

(ii) A sphere is expanding so that its surface area increases at the constant rate of 60 mm2/s. At what rate is its volume increasing when the radius is 20 mm.

36. Gas is escaped from a sherical balloon, find the radius of the sphere at the instant when the rate of decrease of the

(i) radius and volume are equal; (ii) radius and surface area are equal; (iii) volume and surface are are equal;

Also find how fast the surface area is shrinking at the instant when the radius is 8 m, if the gas is then escaping at the rate of 4 m3 /min.

37. Sand is pouring from a chute at the rate of 12 m3/min. The falling sand forms a conical pile on the ground, whose altitude is always equal to the diameter of the base. Show that when the altitude is h m, the volume V m3 of sand in the pile is given by .

Hence determine how fast the altitude is increasing when the pile if 4 m high.

38. A vessel is in the shape of a right circular cone with vertex downwards and axis vertical; the height of the vessel is 12 m and the radius of its circular top is 8 m. The vessel is being filled with water, the inflow of water is at the uniform rate of 2 m3/min.

When the height of water is h m, find the expression for the radius r m of the water surface in terms of h, and prove that the volume V m3 of water then is .

At what rate is the surface of the water rising when its depth is 3 m ?

39. A horizontal trough 10 m long has a cross-section in the shape of a right-angled isosceles triangle. If water is poured in at the rate of 8 m3/min, at what rate is the water level rising when the depth of the water is 2 m ?

40. Water is pouring steadily at the rate of 1 m3/min into a conical reservoir whose semi-vertical angle is 30 degree. When the water is 3 m deep, find the rate at which the

(i) water level is rising;

(ii) area of the water surface is increasing;

(iii) wetted surface area of the reservoir is increasing. (Notice that the curved surface area of a cone is prs, where r is the base radius and s is the slant height).

Answers:

19.(i) 42 (ii) 51 (iii)-24/7 20(i) 3/4 (ii) 27/2 (iii) 1/16 21.(a) 8 cm2/sec (b)sqrt(2)/2 cm/s 22.(a) 600cm3/sec (b) 240cm2/sec (c) 2 cm/s 23.(a) 20 p cm2 (b) 0.8 pcm/s 24. (a) 40 p cm2/s (b) 200pcm3/s 25.(a) 8cm/min (b) 6cm/min 26.(i)5/6p cm/s (ii) 0.0002 cm/s 27. (i)1/25 pcm/s (ii) (a) 64 units/s (b) 36 units/s 28.(i) 5/3 (ii) (7/2, 63/4) 29.5/81 cm/s 30.1.732 cm2/s 31.(i) 1 cm/min (ii) 16 cm2/min (iii) 3 cm/min 32. (i) 2/81pcm/s (ii) 1200 pcm2/min 33. (i) 3/100p m/min (ii) 6m2/min 34.(i) 1/9 cm/s (ii) 12 cm3/s 35.(i) 0.0008 pcm2/s (ii) 600 mm3/s 36.(i) 1/2 (ii) 1/8p (iii) 2; 1 m2/min 37. 3pm/min 38. r=2h/3; 2p m/min 39.1/5 m/min 40. (i)1/3p m/min (ii) 2/3 m2/min (iii) 4/3 m2/min