91Ó°ÊÓ

If you invest \(P\) dollars (the present value of your investment) in a fund that pays an interest rate of \(r\), as a decimal, compounded yearly, then after \(t\) years your investment will have a value \(F\) dollars, which is known as the future value. The discount rate \(D\) for such an investment is given by $$ D=\frac{1}{(1+r)^{t}}, $$ where \(t\) is the life, in years, of the investment. The present value of an investment is the product of the future value and the discount rate. Find a formula that gives the present value in terms of the future value, the interest rate, and the life of the investment.

Show that the following data cannot be modeled by a quadratic function. $$ \begin{array}{|l|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline P(x) & 5 & 8 & 17 & 38 & 77 \\ \hline \end{array} $$

Studies to fit a logistic model to the Eastern Pacific yellowfin tuna population have yielded $$ N=\frac{148}{1+3.6 e^{-2.61 t}}, $$ where \(t\) is measured in years and \(N\) is measured in thousands of tons of fish. \({ }^{7}\) a. What is the \(r\) value for the Eastern Pacific yellowfin tuna? b. What is the carrying capacity \(K\) for the Eastern Pacific yellowfin tuna? c. What is the optimum yield level? d. Use your calculator to graph \(N\) against \(t\). e. At what time was the population growing the most rapidly?