Algebra - Solving exponential & logarithmic functions

Exercise 3.1
25 Continuous Compounding If $8000 is invested for t years at 8% interest compounded continuously, the future value is given by S=8000e0.08t dollars.
a. Graph this function for 0< t < 15.
b. Use the graph to estimate when the future value will be $20,000.

27. Radioactive Decay The amount of radioactive isotope thorium-234 present at time t is given by A (t) =500e¯0.02828t grams, where t is the time in years that the isotope decays. The initial amount present is 500 grams.
a. How many grams remain after 10 years?
b. Graph this function for 0< t < 100.
c. If the half-life is the time it takes for half of the initial amount to decay, use graphical method to estimate the half-life of this isotope.

35. Population The population is a certain city was 53,000 in 2000, and it future size is predicted to be P (t) =53,000e0.015t people, where t is the number of years after 2000.
a. Does this model indicate that the population is increasing or decreasing?
b. Use this function to estimate the population of the city 2005.
c. Use this function to predict the population of the city in 2010.
d. What is the average rate of growth between 2000 and 2010?

37. Carbon-14 Dating An exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. Carbon-14 decays over time, with the amount remaining after t years given by y=100e¯0.00012378t if 100 grams is the original amount.
a. How much remains after 1000 years?
b. Use graphical methods to estimate the number of years until 10 grams of carbon-14 remain.

39. Normal Curve The 'curve' on which many students like to be graded is the bell-shaped normal curve. The equation y=1__ e - (x-50)²/2 describe the normal curve for a standardized
V¯2П¯
test, where x is the test score before curving.
a. Graph this function for x between 47 and 53 and for y between 0 and 0.5.
b. The average score for the test is the score that gives the largest output y. use the graph to find the average score.

Exercise 3.2
43. Life Span On the basic of data for the years 1910 through 1998, the expected life span of people in the United State can be described by the function f(x)=12.734 In x+17.875 years, where x is the number of years from 1900 to the person's birth year.
a. What does this model estimate the life span to be for people born in 1925? In 1996? (Give each answer to the nearest year.)
b. Explain why these number are so different.

51. Suppose the weekly cost for the production of x units of a product is given by
C(x) =3452 + 50 In(x+1) dollars. Use graphical methods to estimate the number of units produced if the total cost is $3556.

Exercise 3.3
53. Snapple Beverage Revenues Prior to the November 1994$1.7 billion takeover proposal by Quaker Oats. Snapple Beverage Corporation's revenues were given by the function
B (t) =1.337e0.718t million dollars, where t is the number of years after 1985.
a. According to the model, what was Snapple's 1995 revenue?
b. If the revenue continued to increase as described by this model, when did it reach $3599 millions?

55. Purchasing Power The purchasing power (real value of money) decreases if inflation is present in the economy. For example, the purchasing power of $40,000 after t year of 5% inflation is given by the model.

P=40,000e¯0.5t dollars

How long will it take value of a $40,000 pension to have a purchasing power of $20,000 under 5% inflation?

61. Doubling Time The number of quarters needed to double an investment when a lump sum is invested at 8% compounded quarters is given by n=log1.02 2.
a. Use the change of base formula to find n.
b. In how many years will the investment double?

67. Radioactive Decay The amount of radioactive isotope thorium-234 percent in a certain sample at time t is given by A(t)=500e¯0.02828t grams, where t years is the time since the initial amount was measured.
a. Find the initial amount of the isotope that percent in the sample.
b. Find the half-life of this isotope. That is, find the number of years until half of the original amount of the isotope remains.

Exercise3.4
17. World Population The following table gives the world population for selected years from 1650 to 2001.
a. Create an exponential function that models these data, with x representing the years after 1600 and y the population in millions. Round the model to four-decimal-place accuracy.
b. Graph the data and the exponential function that model the data on the same axes with window [0,402] by [0, 6500].

Year Population(millions) Year Population(millions)
1650 503 1950 2406
1750 711 1996 5771
1800 913 1999 6000
1850 1131 2001 6200
1900 1590

23. Life Span The table below gives the life expectancy for the people in the United State for the birth years 1910-1998.
a. Find the logarithmic function that models these data, with x equal to 0 in 1900.
b. Find the quadratic function that is the best fit for the date. Round the quadratic coefficient to five decimal places.
c. Graph each of these functions on the same axes with the data points to determine visually which function is the best model for the data for the years 1910-1998.
d. Evaluate both models for the birth year 2010. Which model is better for prediction of life span after 2010?

25. Sexually Active Girls The percent of girls age x or younger who have been sexually active is given in the table below.
a. Create a logarithmic function that models the data, using an input equal to the age of the girls.
b. Use the model to estimate the percent for the girls age 17 or younger who have been sexually active.
c. Find the quadratic function that is the best fit for the data.
d. Graph each of these functions on the same axes with data points to determine which function is the better model for the data.

Age Cumulative percent Sexually Active Girls Cumulative percent Sexually Active Boys
15 5.4 16.6
16 12.6 28.7
17 27.1 47.9
18 44.0 64.0
19 62.9 77.6
20 73.6 83.0

Exercise 3.5
15. Future Value If $8800 is invested for x years at 8% interest compounded annually, find the future value that result in
a. 8 years
b. 30 years

29. Doubling time Use a spreadsheet, a table, or a graph to estimate how long it takes for an investment to double if it is invested at 10% interest.
a. Compounded annually.
b. Compounded continuously.

35. Doubling Time If the money is invested at 10% interest compounded quarterly, the future value of the investment doubles approximately every 7 years.
a. Use this information to complete the table below for an investment of $1000 at 10% interest compounded quarterly.
b. Create an exponential function, around to three decimal places, that models the discrete function defined by the table.
c. Because the interest is compounded quarterly, this model must be interpreted discretely. Use the rounded function to find the value of the investment in 5 years and in 10½ years after the money was invested.
Year 0 7 14 21 28
Future Value($) 1000

Exercise 3.6
9. College Tuition New parent want to put a lump sum into a money market fund to provide $300,000 in 18 years, to help pay for college tuition for their child. If the fund average 10% per year compounded monthly, how many should they invest?

17. Business Sale A man can sell his Thrifty Electronics business for $800,000 cash or for $100,000 plus $122,000 at the end of each year for 9 years.
a. Find the present value of the annuity that is offered if money is worth 10% compounded annually.
b. If he takes the $800,000, spends $100,000 of it, and invests the rest in a 9-year annuity at 10% compounded annually, what size annuity payment will he receive at the end of each year?
c. Which is better, taking the $100,000 and the annuity of taking the cash settlement? Discuss the advantage of your choice.

21. Loan Repayment A loan of $10,000 is to be amortized with quarterly payments over 4 years. If the interest on the loan is 8% per year, paid on the unpaid balance,
a. What is the interest rate charged each quarter on the unpaid balance?
b. How many payments are made to repay the loan?
c. What payment is required each quarterly to amortize the loan?

23. Home Mortgage A couple who wants to purchase a home with a price of $350,000 has $100,000 for a down payment. If they can get a 30-year mortgage at 6% per year on the unpaid balance,
a. What will be their monthly payment?
b. What is the total amount they will pay before they own the house outright?
c. How much interest will they pay over the life of the loan?

Esercise3.7
11. Sexually Active Boys The percent of boys between ages 15 and 20 that been sexually active at some time (the cumulative percent) can be modeled by the logistic function

Y = 89.786_______
1 + 4.6531e - 0.8256x

Where t is the number of years after age 15.
a. Graph this function for 0 < x < 5.
b. What does the model estimate the cumulative percent to be for boys whose age is 16?
c. What cumulative percent does the model estimate for boys of age 21, if it is valid after age 20?
d. What is the limiting value implied by this model?

23. Spread of Disease An employee brings a contagious disease to an office with 150 employees. The number of employees infected by the disease t days after the employees are first exposed to it is given by

N = 100__
1 + 79e¯0.9t

Use graphical or numerical methods to find the number of days until 99 employees have been infected.