Math topic中文

Connected rates of change

Connected rates of change problems.

Practice questions

  1. 1A drilling company plans to install a straight pipeline ABAB through a mountain. Points (x,y,z)(x, y, z) are defined relative…
  2. 2NJC JC1 Promo 9758/2019/Q6 A vessel is formed by removing a smaller cone of radius 5 m from a bigger cone whose semi-ver…
  3. 3[The surface area and volume of a sphere are given by 4πr24\pi r^2 and 43πr3\dfrac{4}{3}\pi r^3 respectively.] An aquarist wh…
  4. 4Fig. 1 shows a piece of card, ABCDABCD, in the form of a rhombus of sides aa cm. The card is made up of 2 identical equil…
  5. 5The diagram shows a V-shaped tank with dimensions L=4 mL = 4\text{ m}, W=0.6 mW = 0.6\text{ m} and H=0.7 mH = 0.7\text{ m}. The tank…
  6. 6In the diagram, point QQ is 400 m400\text{ m} vertically above point AA. Initially, a balloon is at point QQ and moves h…
  7. 7An upright cone, with a closed circular base, is shown in the diagram above. It has a circular base radius of 4 cm4\text{ cm}…
  8. 8In a particular building, there is an LL-shaped corridor (see diagram below). Corridor 1 has a fixed width of 3 m and C…
  9. 9The point PP travels along the curve CC with equation y=xsin⁡−1xy = x \sin^{-1} x, −1<x<1-1 < x < 1. Let the gradient of the curve…
  10. 10Sketch the curve with equation x2400+(y−10)2900=1.\frac{x^2}{400} + \frac{(y-10)^2}{900} = 1. [There is no need to indicate the x-inter…
  11. 11A curve C is defined by the parametric equations x=cos⁡θ−1cos⁡θ,  y=cos⁡θ+1cos⁡θ,  −π2<θ<π2.x = \cos\theta - \dfrac{1}{\cos\theta},\ \ y = \cos\theta + \dfrac{1}{\cos\theta},\ \ -\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}.…
  12. 12A hemispherical goldfish tank with radius 15 cm (as shown in the figure above) was initially filled with water. The tank…
  13. 13NJC JC1 Promo 9758/2019/Q6 A vessel is formed by removing a smaller cone of radius 5 m from a bigger cone whose semi-ver…
  14. 14The diagram shows a right circular cone of height of 2 units, radius rr and slant height ll inscribed in a sphere of r…
  15. 15[A sphere of radius rr has surface area 4πr24\pi r^2 and volume 43πr3\frac{4}{3}\pi r^3.] A water fountain is to be construc…
  16. 16A curve CC has equation 3x2−2xy+5y2=14.3x^2 - 2xy + 5y^2 = 14. Show that dydx=3x−yx−5y.\frac{dy}{dx} = \frac{3x - y}{x - 5y}. Find the exact…
  17. 17[Cambridge/1998] A spherical balloon is being inflated and at the instant when its radius is 3 m, its surface area i…
  18. 18A company manufactures a closed container made of glass as shown in the figure below. The closed container, of negligibl…
  19. 19A hemispherical goldfish tank with radius 15 cm (as shown in the figure above) was initially filled with water. The tank…
  20. 20A popular toy company is designing a new water play feature for children. The toy consists of a cylindrical water contai…
  21. 21In an experiment to test the design theory of launching a new space telescope into space, scientists monitor the movemen…
  22. 22[Cambridge/9709/2009] An underground storage tank is being filled with liquid as shown in the diagram. Initially the…
  23. 23A wafer fabrication company uses the floating-zone method to purify polysilicon ingots, each having a uniform cross-sect…
  24. 24[It is given that a cone with radius rr, height hh and slant height ll has curved surface area πrl\pi r l.] A drinking…
  25. 25[Cambridge/9709/2004] A rectangular reservoir has a horizontal base of area 1000 m2\text{m}^2. At time t=0t = 0, it i…
  26. 26Jim places an 8 m rod XYXY against a vertical wall, as shown in the diagram below. The points O,XO, X and YY are coplana…
  27. 27In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo un…
  28. 28David flies a drone with a flight path CC, modelled by the equation y=2xy = 2^x, where x60;0x 60; 0 and y>0y > 0. He stands a…
  29. 29A cone-shaped cup is made of paper of negligible thickness to hold 20π cm320\pi\text{ cm}^3 of liquid. The open inverted cone…
  30. 30A curve CC has parametric equations x=λ1+λ3andy=λ21+λ3,for λ>0.x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for } \lambda > 0.…
  31. 31[ The volume of a cone of base radius rr and height hh is given by V=13πr2hV = \frac{1}{3}\pi r^2 h. ] A disposable cup for…
  32. 32Fig. 1 shows a piece of card, ABCDABCD, in the form of a rhombus of sides aa cm. The card is made up of 2 identical equil…
  33. 33A metal ball is released from the surface of the liquid in a tall cylinder. The ball falls vertically through the liquid…
  34. 34Water is poured at a rate of 0.1 m3\text{m}^3 per minute into a container in the form of an open cone. The semi-vertical…
  35. 35A circular sector has radius rr cm and angle θ\theta radians. This sector has area A cm2A\text{ cm}^2 and fixed perimeter…
  36. 36Sketch the curve with equation x2400+(y−10)2900=1.\displaystyle\frac{x^2}{400}+\frac{(y-10)^2}{900}=1. [There is no need to indicate th…
  37. 37In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo un…
  38. 38[Cambridge/2006] Find the coordinates of the points on the curve 3x2+xy+y2=333x^2 + xy + y^2 = 33 at which the tangent is pa…
  39. 39[A sphere of radius rr has surface area 4πr24\pi r^2 and volume 43πr3\dfrac{4}{3}\pi r^3.] A water fountain is to be constru…
  40. 40In the diagram, point Q is 400 m vertically above point A. Initially, a balloon is at point Q and moves horizontally in…
  41. 41Jesse leans a 6 metre rod XYXY in the position as shown below. At time tt seconds, the two ends XX and YY are xx and…
  42. 42A particle moving along a path at time tt, where 0<t<π30 < t < \frac{\pi}{3}, is defined parametrically by x=cot⁡3tandy=2csc⁡3t+1.x = \cot 3t \quad \text{and} \quad y = 2\csc 3t + 1.…
  43. 43Fig. 1 A tablet is dissolving in water and is modelled as a cylinder, as shown in Fig. 1. At t seconds after being dropp…
  44. 44A ladder of length 3.12 m is sliding down a vertical wall such that the foot of the ladder is moving along the floor at…
  45. 45A tablet is dissolving in water and is modelled as a cylinder, as shown in Fig. 1. At tt seconds after being dropped in…
  46. 46A particle moving along a path at time tt, where 0<t<π30 < t < \dfrac{\pi}{3}, is defined parametrically by x=cot⁡3tx = \cot 3t a…
  47. 47[The volume of a sphere with radius rr is given by 43πr3\frac{4}{3}\pi r^3 and the surface area of a sphere with radius rr…
  48. 48As part of a game, a person stands xx m away from a wall and tries to throw an object through a 0.40 m wide hole in the…
  49. 49Trinity de Rale is a budding installation artist. Her latest mobile artwork comprises of two fixed rails placed at an an…