HW 3.0.1: Simplify Using Exponent and Logarithmic Properties

Use the properties of Exponents and Logarithms to simplify each expression without a calculator.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32.

33. 34. 35. 36.

37. 38. 39. 40.

41. 42.

Answers: 1. 8 2. 3. 32 4. 9 5. 6. 0.001 7. 1

8. 2 9. 10. -4 11. -1 12. 2 13. -1 14. 1

15. 16. 3 17. 3 18. 5 19. -2 20. 21.

22. -4 23. -3 24. 1 25. -6 26. -2 27. 6 28.

29. 0 30. 31. 32. 3 33. 216 34. 216 35. 5

36. 37. 38. 39. 1 40. 2 41. -1 42. 0

HW 3.0.2: Expand and Condense Logarithms

In Exercises 1 – 15, expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers.

1. 2. 3.

4. 5. 6.

13. 14. 15.

In Exercises 16-26, use the properties of logarithms to write the expression as a single logarithm.

16. 17. 18.

19. 20. 21.

22. 23. 24.

25. 26. Answers:

1. 2. 3.

4. 5. 6.

7. 8.

9. 10. 11.

12. 13.

14. 15.

16. 17. 18.

19. 20. 21.

22. 23. 24.

25. 26.

HW 3.1.2: Compound Interest

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ n Compounds per year Compound Continuously A= Amount P = Principal (initial investment) r = APR (annual percent rate, decimal) t = number of years n = number of compounds per year Do each of the bullets for Exercises 1-6.  Find the amount, , in the account as a function of the term of the investment t in years.

 Determine how much is in the account after 5 years, 10 years, 30 years, and 35 years. Round your answers to the nearest cent.

 Determine how long will it take for the initial investment to double. Round your answer to the nearest year.

1. $500 is invested in an account, which offers 0.75%, compounded monthly.

2. $500 is invested in an account, which offers 0.75%, compounded continuously.

3. $1000 is invested in an account, which offers 1.25%, compounded quarterly.

4. $1000 is invested in an account, which offers 1.25%, compounded continuously.

5. $5000 is invested in an account, which offers 2.125%, compounded daily.

6. $5000 is invested in an account, which offers 2.125%, compounded continuously.

7. Look back at your answers to Exercises 1-6. What can be said about the differences between monthly, daily, or quarterly compounding and continuously compounding the interest in those situations?

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ 8. How much money needs to be invested now to obtain $2000 in 3 years if the interest rate in a savings account is 0.25%, compounded continuously? Round your answer to the nearest cent.

9. How much money needs to be invested now to obtain $5000 in 10 years if the interest rate in a CD is 2.25%, compounded monthly? Round your answer to the nearest cent.

10. On May 31, 2014, the Annual Percentage Rate listed at Jeff’s bank for regular savings accounts was 0.25%, compounded monthly.

a. If what is ?

b. Solve the equation for t.

c. What principal P should be invested so that the account balance is $2000 in three years?

11. Jeff’s bank also offers a 36-month CD (Certificate of Deposit) with an APR of 2.25%

a. If what is ?

b. Solve the equation for t.

c. What principal P should be invested so that the account balance is $2000 in three years?

12. Show that the time it takes for an investment to double in value does not depend on the principal P, but rather, depends only on the APR and the number of compoundings per year. Let and with the help of your classmates compute the doubling time for a variety of rates r. Then look up the Rule of 72 and compare your answers to what the rule says. If you are really

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ interested in Financial Mathematics, you could also compare and contrast the Rule of 72 with the Rule of 70 and the Rule of 69.

Answers: 1.    years for the investment to double. 2.    years for the investment to double. 3.     years for the investment to double. 4.     years for the investment to double. 5.    years for the investment to double.

6.    years for the investment to double.

8. $1985.06 9. $3993.42 10. a. $2040.40 b. 277.29 years c. $1985.06 11. a. $2394.03 b. 30.83 years c. $1869.57

HW 3.2.1: Half-Life or = Initial amount t = time in (unit given as half-life) h = length of half-life

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ In Exercises 1-5, do each of the bullets  Create a function, , for the amount of isotope after t years  Determine how long it takes for 90% of the material to decay. Round answers to two decimal places.

1. Cobalt 60, used in food irradiation, initial amount 50 grams, half-life of 5.37 years.

2. Phosphorus 32, used in agriculture, initial amount 2 milligrams, half-life 14 days.

3. Chromium 51, used to track red blood cells, initial amount 75 milligrams, half-life 27.7 days.

4. Americium 241, used in smoke detectors, initial amount 0.29 micrograms, half-life 432.7 years.

5. Uranium 235, used for nuclear power, initial amount 1 kg grams, half-life 704 million years.

Answers:

1. 17.84 years

2. 46.51 days

3. 92.02 days

4. 1437.40 years

5. 2338.64 million years

HW 3.2.1: Solving Equations using e and ln

Solve the equation analytically, using the property that the natural base and the natural log are inverse, and . Round answers to the nearest thousandth, show work.

1.) 2.) 3.)

4.) 5.) 6.)

7.) 8.) 9.)

13.) 14.) 15.)

16.)

Answers: 1.) 0.00012 2.) 6.931 3.) -0.347

4.) 28.904 5.) -5.108 6.) 0.693

7.) 0.231 8.) 4.209 9.) 0.693

10.) 0.173 11.) 12.) 2.117

13.) 0.0524 14.) e 15.) 0.5

16.) 2.5

HW 3.3.1: Exponential Growth and Decay

For each problem use the form or, where k is a constant that describes the situation and t is time.

A population numbers 11,000 organisms initially and grows by 8.5% every three years. Write an exponential model for the population.

A vehicle is currently $6,000 and has been decreasing by 10.2% every two years. Write an exponential model for the population.

The fox population in a certain region has an annual growth rate of 9 percent per year. It is estimated that the population in the year 2010 was 23,900. Estimate the fox population in the year 2018.

A vehicle purchased for $32,500 depreciates at a constant rate of 5% each year. Determine the approximate value of the vehicle 12 years after purchase.

A business purchases $125,000 of office furniture, which depreciates at a constant rate of 12% every 3 years. Find the residual value of the furniture 10 years after purchase.

Find a formula for an exponential function passing through the two points.

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ 6. 7. 8. Use Substitution to find a formula for an exponential function passing through the two points. 9. 10. 11. 1. A radioactive substance decays exponentially. A scientist begins with 100 milligrams of a radioactive substance. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours?

A car was valued at $38,000 in the year 2003. The value depreciated to $11,000 by the year 2009. Assume that the car value continues to drop by the same percentage. What will the value be in the year 2013?

A house was valued at $110,000 in the year 1985. The value appreciated to $145,000 by the year 2005. What was the annual growth rate between 1985 and 2005? Assume that the house value continues to grow by the same percentage. What did the value equal in the year 2010?

An investment was valued at $11,000 in the year 1995. The value appreciated to $14,000 by the year 2008. What was the annual growth rate between 1995 and 2008? Assume that the value continues to grow by the same percentage. What did the value equal in the year 2012?

Sketch a graph by hand of the following functions and identify as growth or decay, any key coordinates the graph pass through, and any asymptotes. 16. 17. 18. 19. 20. 21.

Answers: 1. 2. 3. 47,622

4. $17,561 5. $81,630.46

6. 7. 8. 9. 10. 11.

12. , 13. , $4,812

14. , Annual Growth Rate = 1.39%,

15. , Annual Growth Rate 1.87%,

17. Growth, Goes through (1,3), Horizontal Asymptote at y = 0

19. Decay, Goes through (0,2), Horizontal Asymptote y = 0

21. Decay, Goes through (0, -1), Horizontal Asymptote y = -4

HW 3.3.2: The Logistic Growth Model

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ A fundamental population growth model in ecology is the logistic model. In one respect, logistic population growth is more realistic than exponential growth because logistic growth is not unbounded. We can write the logistic model as, where P(t) is the population size at time t (assume that time is measured in days), P0 is the initial population size, K is the carrying capacity of the environment, defined as the maximum population size an environment can support, and r is a constant representing the rate of population growth or decay.

1.) The population of Sasquatch in Bigfoot County is modeled by where P(t) is the population of Sasquatch t years after 2010.

(a) Find and interpret P(0).

(b) Find the population of Sasquatch in Bigfoot County in 2013. Round your answer to the nearest Sasquatch.

(c) When will the population of Sasquatch in Bigfoot County reach 60? Round your answer to the nearest year. (Show your work)

(d) Find and interpret the end behavior of the graph of y = P(t). Check your answer using a graphing utility

2.) A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the pack will be modeled by the logistic curve. where t is measure in months.

(a) Estimate the population after 5 months.

(b) After how many months will the population be 500? (Show your work)

3.) On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled by where F(t) is the total number of students infected after t days. The college will cancel classes when 40% or more of the students are infected.

(a) Graph the function and sketch the curve, label an appropriate domain and range.

(b) How many students are infected after 5 days?

4.) Of a group of 200 college men the number N who are taller than x inches is given below:

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ x: 65 66 67 68 69 70 71 72 (6 ft) 73 74 75 76 N: 197 195 184 167 137 101 71 43 25 11 4 2

(a) Construct a scatter plot for this data and explain why a logistic model is appropriate for this data. If you did logistic regression you would see (b) How is the shape of this curve different from that in number 3a?

(d) The tallest 85 men are all grouped together, all of these men are at least ____ inches tall. (round to the nearest tenth of an inch, and show your work)Answers: 1.a.) . There are 29 Sasquatch in Bigfoot County in 2010 b. Sasquatch. c. years. d. As . As time goes by, the Sasquatch Population in Bigfoot County will approach 120. Graphically, has a horizontal asymptote y = 120.

2.a.) b. months

3.a.) 54 b.) 10.1 days

4.c.) 125 men d.) 70.6 inches

HW 3.4.1: Applications of Function Patterns

Use your knowledge of function patterns to answer the questions below. Calculators should be used to help with the calculations but not for regression.

Linear: Quadratic:

Power: Exponential: Logarithm: 1.) A house is bought in an up and coming Texas neighborhood. The table below shows the value of the house, V(t), after owning the house for t years.

t in years V(t) value of house 2 $150,000

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ 4 $225,000 6 $337,500 8 $506,250 a. Determine the price of the home after 12 years.

b. Use the function properties that you learned previously to make a conjecture as to the type of function that models the given data. What type of function models this pattern?

c. Why can’t you use the pattern to find V(15)?

d. Find a particular equation for V(t) and verify that all of the V(t) values in the given table satisfy the equation. (leave your equation exact, avoid decimals in your equation)

e. Use your equation to calculate V(15).

f. In what year (round to the nearest hundredth) will the house be worth 5 million dollars?

2.) A Cessna airplane is taking off on a runway. You start recording the velocity as a function of time slightly after it has started. Below is a table of your results.

t, observing in V(t) in miles per seconds hour 3 80 6 160 9 206.8 12 240

a. Determine the velocity of the plane after 48 seconds. b. Use the function properties that you learned previously to make a conjecture as to the type of function that models the given data. What type of function models this pattern?

c. Why can’t you use the pattern to find V(18)?

David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ d. Find a particular equation for V(t) and verify that all of the V(t) values in the given table satisfy the equation.

e. Use your equation to calculate V(18).

f. How long will it take the plane to reach 700 mph?

3.) Below is a table relating altitude, h, and atmospheric pressure P(h). Altitude above sea Atmospheric Pressure level (km) (kPa) 2 80 4 64 6 51.2 8 40.96

a. Determine the atmospheric pressure at 10 km above sea level.

b. Use the function properties that you learned previously to make a conjecture as to the type of function that models the given data. What type of function models this pattern?

c. Why can’t you use the pattern to find P(5)?

d. Find a particular equation for P(h) and verify that all of the P(h) values in the given table satisfy the equation. (leave your equation exact, avoid decimals in your equation)

e. Use your equation to calculate P(5).

f. At what altitude will the atmospheric pressure reach 25 kPa?

4.) Below is a table relating speeds, s, of a vehicle and its stopping distance, d(s). Highway Safety Division Speed (mph) Stopping David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ Distance (ft) 10 14.6 20 39.9 30 75.2 40 120.5 50 175.8 a. Determine the stopping distance at 70 mph.

b. Use the function properties that you learned previously to make a conjecture as to the type of function that models the given data. What type of function models this pattern?

c. Why can’t you use the pattern to find d(25)?

d. Find a particular equation for d(s) and verify that all of the d(s) values in the given table satisfy the equation.

e. Use your equation to calculate d(25).

f. At what speed will the stopping distance be 100ft? Answers: 1.a.) $1,139,062.50 b.) Exponential: Add-Product c.) not a multiple of 2 (even) d.) or e.) $2,092,591.33 f.) 19.296 years

2.a.) 400 seconds b.) Logarithmic: Product-Add c.) V(18) is not a product 3, 6, 12, 24, 48, 96,… d.) e.) 287.797 mph f.) 645.809 seconds

3.a.) 32.768 kPa b.) Exponential: Add-Product c.) P(5) not a multiple of 2 d.) or e.) 57.243 kPa f.) 12.425 km

4.a.) 316.4 ft b.) Second difference – Quadratic c.) The input is not a multiple of 10 David Lippman and Melonie Rasmussen © 2015. Precalculus: An Investigation of Functions Ed. 1.5. Retrieved from: http://www.opentextbookstore.com/precalc/ You can make the rest of this a bonus and have them do regression, using matrices since this could make a parabola and 3 points are given. d.) e.) 56.3 ft f.) 35.744 seconds