91Ó°ÊÓ

$$\text {Rewrite using rational exponents.}$$ $$\sqrt{5}$$

Evaluate each expression to four decimal places using a calculator. $$3.2^{1 / 2}$$

Simplify the expression. $$\sqrt[3]{2\left(\frac{1}{2} x^{3}+\frac{5}{2}\right)-5}$$

In Exercises \(11-14,\) use \(f(t)=4 e^{t}\) For what value of \(t\) will \(f(t)=8 ?\)

Evaluate each expression to four decimal places using a calculator. $$e^{-3.2}$$

Evaluate each expression without using a calculator. $$\ln e^{w}$$

The half-life of plutonium-238 is 88 years. (a) Given an initial amount of \(A_{0}\) grams of plutonium238 at time \(t=0,\) find an exponential decay model, \(A(t)=A_{0} e^{k t},\) that gives the amount of plutonium238 at time \(t, t \geq 0\). (b) Calculate the time required for \(A_{0}\) grams of plutonium- 238 to decay to \(\frac{1}{3} A_{0}\).

The population of the United States is expected to grow from 282 million in 2000 to 335 million in \(2020 .\) (Source: U.S. Census Bureau) (a) Find a function of the form \(P(t)=C e^{k t}\) that models the population growth of the United States. Here, \(t\) is the number of years since 2000 and \(P(t)\) is in millions. (b) Assuming the trend in part (a) continues, in what year will the population of the United States be 300 million?

Sketch the graph of each function. $$f(x)=3^{-x}+1$$

The spread of a disease can be modcled by a logistic function. For example, in carly 2003 there was an outbreak of an illness called SARS (Severe Acute Respiratory Syndrome) in many parts of the world. The following table gives the total momber of cases in Canada for the wecks following March 20,2003 (Source: World Health Organization) (Note: The total number of cases dropped from 149 to 140 between weeks 3 and 4 because some of the cases thought to be SARS were reclassified as other discases.) $$\begin{array}{|c|c|}\hline\text { Weeks since } & \\\\\text { March } 20,2003 & \text { Total Cases } \\\0 & 9 \\\1 & 62 \\\2 & 132 \\\3 & 149 \\\4 & 140 \\\5 & 216 \\\6 & 245 \\\7 & 252 \\\8 & 250 \\\\\hline\end{array}$$ (a) Explain why a logistic function would suit this data well. (b) Make a scatter plot of the data and find the logistic function of the form \(f(x)=\frac{\epsilon}{1+a \varepsilon^{-1}}\) that best fits the data. (c) What docs \(c\) signify in your model? (d) The World Health Organization declared in July 2003 that SARS no longer posed a threat in Canada. By analyring this data, explain why that would be so.