Bài tập CALCULUS 74

Created by T Madas Question 252 (****+) A D F x C B E The figure above shows the design of a horse feeder which in the shape of a hollow, open topped triangular prism The triangular faces at the two ends of the feeder are isosceles and right angled, so that AB = BC = DE = EF and ABC = DEF = 90° The triangular faces are vertical, and the edges AD , BE and CF are horizontal The capacity of the feeder is m3 a) Show that the surface area, A m , of the feeder is given by A= 16 x + , x where x is the length of AC b) Determine by differentiation the value of x for which A is stationary, giving the answer in the form k , where k is an integer c) Show that the value of x found in part (b) gives the minimum value for A [continues overleaf] Created by T Madas Created by T Madas [continued from overleaf] d) Show, by exact calculations, that the minimum surface area of the feeder is 12 m e) Show further that the length ED is equal to the length EB C2V , x = 2 ≈ 2.82 , ED = EB = Created by T Madas Created by T Madas Question 253 (****+) y M y = − x2 + 8x − A ( 6,5 ) x O The figure above shows the quadratic curve with equation y = − x2 + 8x − The point M is the maximum point of the curve and A is another point on the curve whose coordinates are ( 6,5 ) Find the exact area of the shaded region, bounded by the curve, the x axis and the straight line segment from A to M SYN-M , area = 104 Created by T Madas