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Mathematics In Biology INumbers and Percentages below are fictitious. This assignment explores two different ways of describing growth of populations, one using doubling time and the another using annual growth rates. Doing the following questions will show you how the two different descriptions or models can be interchanged - allow you to switch between doubling times (or halving times) and annual growth or decay rates. 1. The Beluga whale population in the St. Laurent Rivers is decreasing at a rate of 2% per year. So after t = m years, the number left is
where N0 denotes (is, represents) the initial population. (a) Evaluate the factor (0.98)m for m = 0, 5, 10, 15, 20,
25, 30, 35 and 40 years.with the aid of a calculator. 2. (a) For several years, the Blue whale population off an Antarctic is growing at 2.5% per year. At this rate of growth, a population of 1000 would increase as follows.
Fill in the blank population numbers to the nearest whole number to estimate the population, one year after another. (b) Evaluate the formula N(m) = A*(1+i)m for m = 0, 1, 2, 3, 4, 5 and 6, assuming i =2.5% = 0.025 and A = 1000. The results should agree with those computed one year at a time, and one year after another in part (a). (c) Find the number of years m for which the factor (1+i)m has a value equal to 2. Using the numerical or algebraic methods followed earlier in question 1. The algebraic method is better - shows greater mathematical maturity. (d) Let n satisfy (1+i)n = 2. Compute N(p)= A*(1+i)p for p = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%. Do you need to know the value of i if you are given m. (e) Let n satisfy (1+i)n = 2. Compute A*2m/n for m = 0, 1, 2, 3, 4, 5, 6 with A = 1000 again. Use your calculator. (f) Let n satisfy (1+i)n = 2. Compute A*2m/nfor m = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%. 3. The population of ponies on a isolated island doubles every four years for a decade or two. During that period the population numbers N(t) = 300*2m/4when t = m years. Show algebra implies N(t+1) = 2¼ N(t) regardless of the value of t. (a) Find a number i so that 21/4 =
1+i. 4. The population of seagulls on a isolated island halves every four years for a decade due to a harsh environment change. During that period the population numbers N(t) = 300*(½)m/4when t = m years. Show algebra implies N(t+1) = (½)¼ N(t) regardless of the value of t. (a) Find a number i so that (½)1/4 =
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