Bài tập Toán DIFFERENTIATION OPTIMIZATION 03

Created by T Madas Question (***) A 8x 6x E B y D 10 x C The figure above shows a pentagon ABCDE whose measurements, in cm , are given in terms of x and y a) If the perimeter of the pentagon is 120 cm , show clearly that its area, A cm , is given by A = 600 x − 96 x b) Use a method based on differentiation to calculate the maximum value for A , fully justifying the fact that it is indeed the maximum value Amax = 937.5 Created by T Madas Created by T Madas Question (***) x A y B G F 1c E x y C D The figure above shows a clothes design consisting of two identical rectangles attached to each of the straight sides of a circular sector of radius x cm The rectangles measure x cm by y cm and the circular sector subtends an angle of one radian at the centre The perimeter of the design is 40 cm a) Show that the area of the design, A cm , is given by A = 20 x − x b) Determine by differentiation the value of x for which A is stationary c) Show that the value of x found in part (b) gives the maximum value for A d) Find the maximum area of the design x = 10 , Amax = 100 Created by T Madas Created by T Madas Question (***+) r h The figure above shows a closed cylindrical can of radius r cm and height h cm a) Given that the surface area of the can is 192π cm , show that the volume of the can, V cm3 , is given by V = 96π r − π r b) Find the value of r for which V is stationary c) Justify that the value of r found in part (b) gives the maximum value for V d) Calculate the maximum value of V r = ≈ 5.66 , Vmax = 256π ≈ 1137 Created by T Madas ... is 40 cm a) Show that the area of the design, A cm , is given by A = 20 x − x b) Determine by differentiation the value of x for which A is stationary c) Show that the value of x found in part