39 Turn over Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 7 marks) 40 13. r m r m h m Figure 9 [A sphere of radius r has volume 4 3 ʌU࣠3 and surface area 4ʌU࣠2] A manufacturer produces a storage tank. The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness. The cylinder has radius r metres and height h metres and the hemisphere has radius r metres. The volume of the tank is 6 m3. (a) Show that, according to the model, the surface area of the tank, in m2, is given by 12 r + 5 3 ʌU࣠2 (4) The manufacturer needs to minimise the surface area of the tank. (b) Use calculus to find the radius of the tank for which the surface area is a minimum. (4) (c) Calculate the minimum surface area of the tank, giving your answer to the nearest integer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________