91Ó°ÊÓ

At the beginning of an experiment, a culture contains 200 H. pylori bacteria. An hour later there are 205 bacteria. Assuming that the \(H\). pylori bacteria grow exponentially, how many will there be after 10 hours? After 2 days?

(a) Graph \(f(x)=x^{5}\) and explain why this function has an inverse function. (b) Show algebraically that the inverse function is \(g(x)=x^{1 / 5}\) (c) Does \(f(x)=x^{6}\) have an inverse function? Why or why not?

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The probability \(P\) percent of having an accident while driving a car is related to the alcohol level of the driver's blood by the formula \(P=e^{k t},\) where \(k\) is a constant. Accident statistics show that the probability of an accident is \(25 \%\) when the blood alcohol level is \(t=.15\). (a) Find \(k .\) IUse \(P=25,\) not .25 .1 (b) At what blood alcohol level is the probability of having an accident \(50 \% ?\)

Kerosene is passed through a pipe filled with clay to remove various pollutants. Each foot of pipe removes \(25 \%\) of the pollutants. (a) Write the rule of a function that gives the percentage of pollutants remaining in the kerosene after it has passed through \(x\) feet of pipe. [See Example 7.] (b) How many feet of pipe are needed to ensure that \(90 \%\) of the pollutants have been removed from the kerosene?

If inflation runs at a steady \(3 \%\) per year, then the amount a dollar is worth decreases by \(3 \%\) each year. (a) Write the rule of a function that gives the value of a dollar in year \(x .\) (b) How much will the dollar be worth in 5 years? In 10 years? (c) How many years will it take before today's dollar is worth only a dime?

(a) The half-life of radium is 1620 years. If you start with 100 milligrams of radium, what is the rule of the function that gives the amount remaining after \(t\) years? (b) How much radium is left after 800 years? After 1600 years? After 3200 years?

If \(f(x)=A \ln x+B\) and \(f(e)=5\) and \(f\left(e^{2}\right)=8,\) what are \(A\) and \(B ?\)

The beaver population near a certain lake in year \(t\) is approximately $$p(t)=\frac{2000}{1+199 e^{-.5544 t}}$$ (a) When will the beaver population reach \(1000 ?\) (b) Will the population ever reach \(2000 ?\) Why?

In the year 2009 , Olivia's bank balance is 1000 dollars. In the year 2010 , her balance is 1100 dollars. (a) If her balance is growing exponentially, in what year will it reach 2500 dollars? (b) If her balance is instead growing linearly, in what year will it reach 2500 dollars ?