91Ó°ÊÓ

Deal with radioactive decay and the function \(M(x)=c\left(.5^{x / h}\right) \). A sample of 300 grams of uranium decays to 200 grams in .26 billion years. Find the half-life of uranium.

The pressure of the atmosphere \(p(x)\) (in pounds per square inch) is given by $$ p(x)=k e^{-0000425 x} $$ where \(x\) is the height above sea level (in feet) and \(k\) is a constant. (a) Use the fact that the pressure at sea level is 15 pounds per square inch to find \(k\) (b) What is the pressure at 5000 feet? (c) If you were in a spaceship at an altitude of 160,000 feet, what would the pressure be?

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$k(x)=\ln (x+2)$$

Find the difference quotient of the given function. Then rationalize its numerator and simplify. $$g(x)=\sqrt{x^{2}-x}$$

A weekly census of the tree-frog population in Frog Hollow State Park produces the following results. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Week } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Population } & 18 & 54 & 162 & 486 & 1458 & 4374 \\ \hline \end{array}$$ (a) Find a function of the form \(f(x)=P a^{x}\) that describes the frog population at time \(x\) weeks. (b) What is the growth factor in this situation (that is, by what number must this week's population be multiplied to obtain next week's population)? (c) Each tree frog requires 10 square feet of space and the park has an area of 6.2 square miles. Will the space required by the frog population exceed the size of the park in 12 weeks? In 14 weeks? [Remember: 1 square mile \(\left.=5280^{2} \text { square feet. }\right]\)

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). At what annual rate of interest should 1000 dollars be invested so that it will double in 10 years if interest is compounded quarterly?

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). How long does it take 500 dollars to triple if it is invested at \(6 \%\) compounded: (a) annually, (b) quarterly, (c) daily?

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). (a) How long will it take to triple your money if you invest 500 dollars at a rate of \(5 \%\) per year compounded annually? (b) How long will it take at \(5 \%\) compounded quarterly?

Find a viewing window (or windows) that shows a complete graph of the function. $$f(x)=\frac{\log x}{x}$$

Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The amount \(P\) of ozone in the atmosphere is currently decaying exponentially each year at a continuous rate of \(\frac{1}{4} \%\) (that is, \(k=-.0025\) ). How long will it take for half the ozone to disappear (that is, when will the amount be \(P / 2\) )? [Your answer is the half-life of ozone.]