Biomass Growth Suppose that you grow plants in several study plots and wish to measure

Chapter 6, Problem 2

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Biomass Growth Suppose that you grow plants in several study plots and wish to measure the response of total biomass to the treatment in each plot. One way to measure this response would be to determine the average specific growth rate of the biomass for each plot over the course of the growing season. We denote by B(t) the biomass in a given plot at time t. Then the specific growth rate of the biomass at time t is given by 1 B(t) dB dt (a) Explain why 1 t % t 0 1 B(s) dB(s) ds ds is a way to express the average specific growth rate over the interval [0, t]. (b) Use the chain rule to show that 1 B(t) dB dt = d dt (ln B(t)) (c) Use the results in (a) and (b) to show that the average specific growth rate of B(s) over the interval [0, t] is given by 1 t % t 0 d ds (ln(B(s)) ds = 1 t ln B(t) B(0) provided that B(s) > 0 for s [0, t]. (d) Explain the measurements that you would need to take if you wanted to determine the average specific growth rate of biomass in a given plot over the interval [0, t]. (Adapted from Herschy, 1995) The speed of water in a channel varies considerably with depth. Due to friction, the speed reaches zero at the bottom and along the sides of the channel. The speed is greatest near the surface of the stream. To find the average speed for the vertical speed profile, two methods are frequently employed in practice: 1. The 0.6 depth method: The speed is measured at 0.6 of the depth from the surface, and this value is taken as the average speed. 2. The 0.2 and 0.8 depth method: The speed is measured at 0.2 and 0.8 of the depth from the surface, and the average of the two readings is taken as the average speed. The theoretical speed distribution of water flowing in an open channel is given approximately by v(d) = _ D d a _1/c (6.19) where v(d) is the speed at depth d below the water surface, c is a constant varying from 5 for coarse beds to 7 for smooth beds, D is the total depth of the channel, and a is a constant that is equal to the distance above the bottom of the channel at which the speed has unit value.