Exam 6: Exponential and Logarithmic Functions

The following figure, from Investor's Business Daily (March 5, 1998), shows how quickly the U.S. metal processing industry isconsolidating. The linear equation that is the best fit for the number of metal processors as a function of years after 1990 is , and the best exponential fit is . The linear equation seems to give a much better fit for the data points than the exponential equation. Why then is the exponential equation a more useful model to predict the number of metal processors in ? ​

The demand function for a certain commodity is given by , where q is number of units. If the price is $1.11 per unit, how many units will be demanded, to the nearest unit? ​

The following table gives the percent of the U.S. population with Internet connections for the years 1997 to 2003. Use a calculator to find the logistic function that models these data. Use x as the number of years past 1995. ​ ​ Source: U.S. Department of Commerce ​

Write the equation in logarithmic form. ​ ​

The president of a company predicts that sales N will increase after she assumes office and that the number of monthly sales will follow the curve given by , where t represents time in months since she assumed office. What is the expected upper limit on sales? ​

Solve the exponential equation. Give the answer correct to 3 decimal places. ​ ​

The demand function for a certain commodity is given by , where q is number of units. At what price per unit will the quantity demanded equal 4 units? Round your answer to the nearest cent. ​

Graph the function. ​ ​

Solve the exponential equation. Round your answer to three decimal places, if necessary. ​ ​

An initial amount of 300 g of the radioactive isotope thorium-234 decays according to , where t is in years. How long does it take for half of the initial amount to disintegrate? This time is called the half-life of this isotope. Round your answer to one decimal place. ​

The following table gives the percent of the U.S. population with Internet connections for the years 1997 to 2003. Use a calculator to find the logistic function that models these data and then use the model to predict the percent of the U.S. population with Internet connections in 2015. Use x as the number of years past 1995. Round your answer to one decimal place. ​ ​ Source: U.S. Department of Commerce ​

Use the fact that the loudness of sound (in decibels) perceived by the human ear depends on intensity levels according to , where is the threshold of hearing for the average human ear. Find the loudness when is 10,000 times . ​

The concentration y of a certain drug in the bloodstream t hours after an oral dosage (with ) is given by the equation . What is the concentration after 6 hours? Round your answer to one decimal place. ​

Use a calculator to evaluate the expression. Round your answer to four decimal places, if necessary. ​ ​

If the demand function for a product is given by where p is the price per unit when x units are demanded, what is the total revenue when 90 units are demanded and supplied? Round your answer to the nearest cent. ​

We will show in the next chapter that if $P is invested for n years at an annual rate of 9% compounded continuously, the future value of the investment is given by ​ ​ Use and graph this function for . ​

If $800 is invested for x years at an annual rate of 9%, compounded quarterly, the future value that will result is ​ ​ Determine the value of the investment after 5 years. Round your answer to two decimal places, if necessary. ​

Total personal income in the United States (in billions of dollars) for selected years from 1960 to 2002 is given in the following table. ​ Source: Bureau of Economic Analysis, U.S. Department of Commerce These data can be modeled by an exponential function. Write the equation of this function, with x as the number of years past 1960. If it's necessary round your calculations to four decimal places. ​

The consumer price index (CPI) is calculated by averaging the prices of various items after assigning a weight to each item. The following table gives the consumer price indexes for selected years from 1940 through 2002, reflecting buying patterns of all urban consumers. Find an equation that models these data and use it to determine when the consumer price index will pass 255. ​ Source: U.S. Bureau of the Census ​

Solve the exponential equation. Give the answer correct to 3 decimal places. ​ ​