Question

2. Yield from an Apple Orchard Figure 6.44 is a graph of the annual yield, y(t), in bushels per year, from an orchardt years after planting. The trees take about 10 years to get established, but for the next 20 years they give a substantial yield. After about 30 years, however, age and disease start to take their toll, and the annual yield falls off. (a) Represent on a sketch of Figure 6.44 the total yield, F(M), up to M years, with 0 <M S 60. Write an expression for F(M) in terms of y(t). (b) Sketch a graph of F(M) against M for 0 SM < 60. (c) Write an expression for the average annual yield, a(M), up to M years. (d) When should the orchard be cut down and replanted? Assume that we want to maximize average revenue per year, and that fruit prices remain constant, so that this is achieved by maximizing average annual yield. Use the graph of y(t) to estimate the time at which the average annual yield is a maximum. Explain your answer geometrically and symbolically.

Answer 1

ylt) → Bushels per year. As we can see, it is a parabola whose vertex is = (33,500). writing general equation of parabola. cy - 500) = -K (X-33)2 (Given parabola is of the general form x?-yay] At X=60, y=100 =) - 400 = -K (27)2 = K = 0.5487 5000.55(4-335) K20.55 (a) Total yield FCM) upto M years. 500 M sot Total yield FCM) is fet). Hence the shaded Expression: FIM) = the sum of annual yield. region represents FCM). (fit) dt. (b) : FIM) is increase, just cumulative, the rate it will continuously of increase will vary

:60 3 33 (c) Average annual yield acm) - FIM) or M ₂ in S fH) at M (d) Average annual yield is maximum around t= 35 years.. observed range of good yield is 20-45 years. Thus after first plarsling, we obtain yield upto 45 years and then replant every 30 years. This can be done in sections of the field as new plants will take 10 years to reach their full yield capacity. So, At to orchard fully planted += 10-40 → max. yield obtained. Here around t= 35 → half orchid replanted so that by t= 45 → it achieves full yield capacity at t= 45 - Remaining half replanted. Thus, this cycle can ensure maximum profits.