Gompertz Growth Curve Consider the function. Q(t) = Ce-Ac**t where Q(t) is the size of a quantity at time t, and A, C, and k are positive constants. The graph of this function, called the Gompertz growth curve, is used by biologists to describe restricted population growth. (a) Determine whether Q(t) is increasing or decreasing over (0, 0). O increasing decreasing (b) Find the time t at which the growth rate Q'(t) is increasing most rapidly. (Hint: Find the inflection point of Q.) (c) Find the limit of Q(t) as t→ ∞.

Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.

Marine Fishery One class of models for population growth rates in marine fisheries assumes that the harvest from fishing is proportional to the population size. For one such model, we have G=0.3n(1n2)0.1n Here G is the growth rate of the population, in millions of tons of fish per year, and n is the population size, in millions of tons of fish. a.Make a graph of G versus n. include values of n up to 1.5 million tons. b.Use functional notation to express the growth rate if the population size is 0.24 million tons, and then calculate that value. c. Calculate G1.42 and explain in practical terms what your answer means. d.At what population size is the growth rate the largest?

Population The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.