91Ó°ÊÓ

The following exponential functions represent population growth. Identify the initial population and the growth factor. a. \(Q=275 \cdot 3^{T}\) b. \(P=15,000 \cdot 1.04^{t}\) c. \(y=\left(6 \cdot 10^{8}\right) \cdot 5^{x}\) d. \(A=25(1.18)^{t}\) e. \(P(t)=8000(2.718)^{t}\) f. \(f(x)=4 \cdot 10^{5}(2.5)^{x}\)

Identify and interpret the decay factor for each of the following functions: a. \(P=450(0.43)^{t}\) b. \(f(t)=3500(0.95)^{t}\) c. \(y=21(3)^{-x}\)

Identify the doubling time or half-life of each of the following exponential functions. Assume \(t\) is in years. [Hint: What value of \(t\) would give you a growth (or decay) factor of 2 (or \(1 / 2\) )?] a. \(Q=70(2)^{t}\) b. \(Q=1000(2)^{t / 50}\) c. \(Q=300\left(\frac{1}{2}\right)^{t}\) d. \(Q=100\left(\frac{1}{2}\right)^{t / 250}\) e. \(N=550(2)^{t / 10}\) f. \(N=50\left(\frac{1}{2}\right)^{t / 20}\)

Given the following exponential decay functions, identify the decay rate in percentage form. a. \(Q=400(0.95)^{t}\) b. \(A=600(0.82)^{\mathrm{r}}\) c. \(P=70,000(0.45)^{t}\) d. \(y=200(0.655)^{x}\) e. \(A=10(0.996)^{T}\) f. \(N=82(0.725)^{T}\)

What is the growth or decay factor for each given time period? a. Weight increases by \(0.2 \%\) every 5 days. b. Mass decreases by \(6.3 \%\) every year. c. Population increases \(23 \%\) per decade. d. Profit increases \(300 \%\) per year. e. Blood alcohol level decreases \(35 \%\) per hour.

Determine which of the following functions are exponential. For each exponential function, identify the growth or decay factor and the vertical intercept. a. \(y=5\left(x^{2}\right)\) b. \(y=100 \cdot 2^{-x}\) c. \(P=1000(0.999)\)

Find \(C\) and \(a\) such that the function \(f(x)=C a^{x}\) satisfies the given conditions. a. \(f(0)=6\) and for each unit increase in \(x,\) the output is multiplied by 1.2 . b. \(f(0)=10\) and for each unit increase in \(x,\) the output is multiplied by 2.5

Lead- 206 is not radioactive, so it does not spontaneously decay into lighter elements. Radioactive elements heavier than lead undergo a series of decays, each time changing from a heavier element into a lighter or more stable one. Eventually, the element decays into lead- 206 and the process stops. So, over billions of years, the amount of lead in the universe has increased because of the decay of numerous radioactive elements produced by supernova explosions. Radioactive uranium- 238 decays sequentially into thirteen other lighter elements until it stabilizes at lead-206. The half-lives of the fifteen different elements in this decay chain vary from 0.000164 seconds (from polonium- 214 to lead- 210 ) all the way up to 4.47 billion years (from uranium- 238 to thorium- 234 ). a. Find the decay rate per billion years for uranium- 238 to decay into thorium- 234 . b. Find the decay rate per second for polonium-214 to decay into lead-2.10.

Each table has values representing either linear or exponential functions. Find the equation for each function. $$ \begin{array}{cccccc} \hline x & -2 & -1 & 0 & 1 & 2 \\ h(x) & 160 & 180 & 200 & 220 & 240 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{cccccc} \hline x & 0 & 10 & 20 & 30 & 40 \\ j(x) & 200 & 230 & 264.5 & 304.17 & 349.8 \\ \hline \end{array} \end{aligned} $$

(Graphing program recommended.) Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide (via plants). As long as a plant or animal is alive, carbon-14 is maintained in the organism at a constant level. Once the organism dies, however, carbon-14 decays exponentially into carbon-12. By comparing the amount of carbon- 14 to the amount of carbon-12, one can determine approximately how long ago the organism died. Willard Libby won a Nobel Prize for developing this technique for use in dating archaeological specimens. The half-life of carbon-14 is about 5730 years. In answering the following questions, assume that the initial quantity of carbon- 14 is 500 milligrams. a. Construct an exponential function that describes the relationship between \(A,\) the amount of carbon- 14 in milligrams, and \(t,\) the number of 5730 -year time periods. b. Generate a table of values and plot the function. Choose a reasonable set of values for the domain. Remember that the objects we are dating may be up to 50,000 years old. c. From your graph or table, estimate how many milligrams are left after 15,000 years and after 45,000 years. d. Now construct an exponential function that describes the relationship between \(A\) and \(T,\) where \(T\) is measured in years. What is the annual decay factor? The annual decay rate? e. Use your function in part (d) to calculate the number of milligrams that would be left after 15,000 years and after 45,000 years.