RI Differentiation C6B Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ___________________________________ Tutorial 6B: Applications of Differentiation Page 1 of 6 Tutorial 6B: Applications of Differentiation Section A (Basic Questions) 1 A curve C has parametric equations 2 4xt t , 32 .yt t The tangent to the curve at the point P where 2t  is denoted by l. (i) Find the cartesian equation of l. [3] (ii) The tangent l meets C again at the point Q. Use a non-calculator method to find the coordinates of Q. [4] [(ii) 21 2yx (iii) (3 ,1 8 ) ] 2 A closed container is constructed us ing a sheet of metal with area 100  cm 2. The container comprises 2 shapes, a cone and a cylinder. The slant height, l, of the cone is 10 cm. Given that the cylinder has height h, and radius r, and the height of the cone is kh, where k is a positive constant. (i) show that 2100 10 2 rrh r  , [1] (ii) use differentiation to show that the exact maximum volume of the container is given that 1 2500132 7Vk  cm3, proving that it is a maximum. [6] [Volume of Cone = 21 3 rh , Curved Surface Area of Cone = rl ] 3 A curve C has equation 32 2 23 3 0 .yx yx   (i) Find d d y x in terms of x and y. [2] (ii) Find the equation of the normal to the curve at the point P(2, 3). [2] (iii) Given that C meets the y-axis at the point R and the normal in (ii) meets the y-axis at the point A, find the area of triangle APR in the form 3ab , where a and b are integers to be determined. [3] [(i) 2 2 d26 d3 4 yy x xy x y   (ii) 1 2 4yx  (iii) a = 4, b = 3]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________ Tutorial 6B: Applications of Differentiation Page 2 of 6 Section B (Discussion Questions) 1 A curve C has equation 22eeyxkx k y k  , where k is a positive constant. (i) Express d d y x in terms of ,xy and k . [3] (ii) Explain why there is no point on C where the tangent is parallel to the x-axis. [2] [(i) de e d2 e xy y yk k xy k x   ] 2 (i) Given that 22 24 0xy x y   , find d d y x in terms of x and y . [4] (ii) For the curve with equation 22 24 0xy x y   , find the coordinates of each point at which the tangent is parallel to the x-axis. [4] [(i) d d yx y xy x   (ii)  2, 2 and 2, 2 ] 3 A designer wishes to create a piece of artwork with painted area of 21200 cm on a rectangular piece of canvas. The painted area measures x cm by y cm and is surrounded by an unpainted border with top and bottom marg ins of 3 cm each, and side margins of 4 cm each on the canvas, as shown in the diagram below. (i) By differentiation, find the dimensions of the canvas with the smallest area. [6] (ii) What is the largest possible area of the canvas if 30 50x ? [2] At an exhibition, a spotlight illuminates a circular region of radius 2  cm on the artwork. The area of this circular region then increases at a constant rate of 220 cm per minute. (iii) Find the rate of change of the radius after 3 minutes. [4] [(i) 48cm by 36cm (ii) 21748 cm (iii) 0.705 cm/min] 4 A curve C has parametric equations etx , lnyt t , where 0t  . (i) Show that there is no tangent to C parallel to the y-axis and find the equation of the tangent to C that is parallel to the x-axis. [5] (ii) Describe the behavior of the tangent to C as 0t . [1] [(i) 1y ] x y 3 3 4 4

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________ Tutorial 6B: Applications of Differentiation Page 3 of 6 5 In the figure (not drawn to scale), POQ is a rail and .6POQ  AB is a rod of fixed length 1 m which is free to slide on the rail with end A on OP and end B on OQ. It is given that OA = x m, OB = y m and OAB  radians. (i) Express x and y in terms of . [2] (ii) Given that S is the area of triangle OAB, find S in terms of . [1] Given that B is moving towards O at a rate of 0.2 ms1 at the instant when 4   , (iii) find the rate of change of S at this instant. [4] [(i) 52sin 6x    , 2siny  (ii) 5sin sin6S    (iii) 0.122 ms1] 6 Water is poured at a constant rate of 20 cm 3 per second into a cup which is shaped like a truncated cone as shown in the figure. The upper and lower radii of the cup are 4 cm and 2 cm respectively. The height of the cup is 6 cm. (i) Show that the volume of water inside the cup, V cm3 is related to the height of the water level, h cm by the equation 3(6 ) 827Vh    . (ii) How fast will the water level be rising when h is 3 cm? Express your answer in exact form. [(ii) -120 cms9 ] y x 1 O Q P B A 

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________ Tutorial 6B: Applications of Differentiation Page 4 of 6 7 A curve C is defined by the parametric equations tan , secxy  for 0 2  . (i) Find the cartesian equation of .C The tangent and normal at tan , secP  meets the x-axis at Q and R respectively. (ii) Show that the area, A of the circle passing through ,PQ and R can be expressed as 2 1tan . 2t a nA     (iii) Show that 2 1 202t t   for all ,0 .tt Deduce the minimum value of A . [(i) 22 1xy (iii) 2 ] 8 A closed cylindrical can with radius r cm and height h cm has fixed volume 20 cm3. The material for the top and bot tom faces costs $0.50 per cm 2 and the material for the curved surface costs $0.30 per cm 2. It also costs $0.80 per cm to weld the top and bottom faces onto the cylinder and $0.60 per cm to weld the seam up the curved surface of the cylinder (see diagram). (i) The total cost of the can is $ C. Show that 2 2 12 123.2Cr r rr   . [3] (ii) Use differentiation to find the values of r and h which give a minimum value of C, proving that C is a minimum. State this value of C. [7] (iii) It is given instead that 0.5 2 r . Find the corresponding range of values of C. [2] [(ii) 1.61r  , 7.68h , $52.37 (iii) 52.37 129.21C ] r h Welding

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ ________________________________ Tutorial 6B: Applications of Differentiation Page 5 of 6 9 A straight street of fixed width a metres on the horizontal ground is bounded on its parallel sides by two vertical walls, one of height 10 metres, and the other of height 9 metres. The intensity of light at point R at ground level on the street is proportional to the angle  , in radians, where PRQ  as shown in the diagram. It is given that x is the distance of R from the base of the wall of height 10 metres. (i) Find an exact expression for  in terms of x, a and  . [1] (ii) Show that  22 d1 0 9 d 100 81xx ax     . Find, in terms of a, the value of x when the intensity of light at R is maximum. [6] (iii) The point R moves across the street towards the wall of height 10 metres at a speed of 0.5 ms-1. Given that 20a , find the rate of change of  at the instant when R is at the midpoint of the street. Give your answer correct to 4 significant figures. [2] [(i) 1110 9tan tanxa x    (ii)  210 3 10 1xa a  (iii) 0.0001381 rad s-1] 10 The diagram below shows the cross-section ABCD of a gutter made using a sheet of metal 25cm wide. The sheet of metal is bent such that