Exam 5: Inverse, Exponential, and Logarithmic Functions

If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula y = y _o e ^-0.3937t , where t is the time in years and y _o is the pollutant level at which further pollution ceased. How many years would it take to clear 59% of the pollutants?

The air pressure p (h) (in lb/ in ² ) at an altitude of h feet above sea level may be approximated by the formula At approximately what altitude h is the air pressure 10 lb/ in ² ?

Graph f on the given interval. Estimate the largest interval [ a , b ] with a < 0 < b on which f is one-to-one.

One method of estimating the thickness of the ozone layer is to use the formula , where I _o is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength. Suppose a wavelength of cm with k = 0.43 , is measured as 1.17. Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

In 1980, the population of blue whales in the southern hemisphere was thought to number 5,000. The population N(t) has been decreasing according to the formula where t is in years and t = 0 corresponds to 1980. Predict the population in the year 2015 if this trend continues.

If monthly payments p are deposited in a savings account paying an annual interest rate r, then the amount A in the account after n years is given by: Estimate n for p = 98, r = 0.02 and A = $208,744.

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

Solve the equation.

Graph f on the given interval. Estimate the largest interval [ a , b ] with a < 0 < b on which f is one-to-one.

Sketch the graph of

How much money, invested at an interest rate of 11% per year compounded continuously, will amount to $105,000 after 15 years?

Express in terms of logarithms of x, y, or z.

Solve the equation without using a calculator.

Change to exponential form.

Find an exponential function of the form f (x) = ba ^x that has the given y-intercept and passes through the point P: y-intercept = 27; P (3,1);

Use the change of base formula to approximate the x-intercept.

Solve the equation.

Simplify the expression.

In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish N( t ) still alive after t years is given by an exponential function. For Pacific halibut, , where N _o is the initial size of the cohort. Approximate the percentage of the original number still alive after 7 years.

Find the exact solution, using common logarithms, and an approximation of the solution of the equation to two decimal places.