The exercise deal with the Shannon index of diversity. Note that in two communities with the same number of species, a larger index indicates greater diversity.

A study of barn owl prey in a particular area produced the following data:

Species	Number
Rats	662
Mice	907
Birds	531

Find the index of diversity of this community. Is this community more or less diverse than the one in the previous exercise?

For the following exercise, do part (a) and skip part (b) if you do not have a graphing calculator. If you have a graphing calculator, then skip part (a) and do part (b).

The table shows the approximate digital share (in percent) of recorded music sales in the United States in recent years.

(a) Let x = 1 correspond to the year 2006. Use the data points for 2006 and 2012 to find a function of the form f(x) = a+b ln x that models these data. [Hint: Use (1,18) to find a; then use (7,59) to find b.

(b) Use logarithmic regression to find a function g that models these data, with x = 1 corresponding to the year 2006.

(c) Assume the model remains accurate and predict the percentage for 2014.

(d) According to the model, when will the share of digital music reach 70%?

Work the following problem.

The number of centenarians (in thousands) in the United States can be approximated by the function h(x) = 2(1.056)$^x$, where x = 0 corresponds to the year 1950. Find the year in which the number of centenarians reached the following levels.

(a) 30,000

(b) 50,000

The barometric pressure p (in inches of mercury) is related to the height h above sea level (in miles) by the equation

h = -5 ln p/29.92

The pressure readings given in parts (a)-(c) were made by a weather balloon. At what heights were they made?

(a) 29.92 inches

(b) 20.05 inches

(c) 11.92 inches

(d) Use a graphing calculator to determine the pressure at a height of 3 miles.

Work the following problem.

Gambling revenues (in billions of /U.S. dollars) generated in Macau, China can be approximated by the function g(x) = 1.104(1.346)$^x$, where x = 2 corresponds to the year 2002. Assume the model remains accurate and find the years in which the gambling revenues reached the indicated amounts.

(a) $20 billion

(b)$45 billion

The average annual expenditure (in dollars) for a consumer of gasoline and motor oil can be approximated by the function

g(x) = -771.9+1035 ln x

where x = 5 corresponds to the year 1995.

(a) Estimate the average expenditure in 2008.

(b) Graph the function g for the period 1995 to 2011.

(c) Assuming the graph remains accurate, what does the shape of the graph suggest regarding the average expenditure on gasoline and motor oil?

For the following exercise, do part (a) and skip part (b) if you do not have a graphing calculator. If you have a graphing calculator, then skip part (a) and do part (b).

The approximate average times (in minutes per month) spent on Facebook per unique visitor in the United States on a mobile device are given in the following table for the last six months of 2012 and first two months of 2013.

(a) Let x = 0 correspond to July, 2012. Use the data points from July, 2012 and February, 2013 to find a function of the form f(x) = a(b$^x$) that models these data.

(b) Use exponential regression to find a function g that models these data, with x = 0 corresponding to July, 2012.

(c) Assume the model remains accurate and estimate the minutes per month in April, 2013.

(d) According to this model, when will the minutes per month reach 1000?

Two people with the flu visited a college campus. The number of days, T, that it took for the flu virus to infect n people is given by

T = -1.43 ln 10,000-n/4998 n

How many days will it take for the virus to infect

(a) 500 people?

(b) 5000 people?

Explain why the equation log (-x) = -4 does have a solution, and find that solution.

The doubling function

D(r) = ln 2/ln (1+r)

gives the number of years required to double your money when it is invested at interest rate r (expressed as a decimal), compounded annually. How long does it take to double your money at each of the following rates?

(a) 4%

(b) 8%

(c) 18%

(d) 36%

(e) Round each of your answers in parts (a)-(d) to the nearest year, and compare them with these numbers: 72/4, 72/8, 72/18, and 72/36. Use this evidence to state a "rule of thumb" for determining approximate doubling time without employing the function D. This rule, which has long been used by bankers, is called the rule of 72.

Explain why the equation $4^{x^2+1}$ = 2 has no solutions.

Another form of Newton's law of cooling is F(t) = $T_0$+Ce$^{-k t}$, where C and k are constants.

A piece of metal is heated to 300 degrees Celsius and then placed in a cooling liquid at 50 degrees Celsius. After 4 minutes, the metal has cooled to 175 degrees Celsius. Find its temperature after 12 minutes.

Work the following problem.

One earthquake measures 4.6 on the Richter scale. A second earthquake has ground motion 1000 times greater than the first. What does the second one measure on the Richter scale?

Solve the following equation.

5(e$^{2 x}$-2) = 15