College Algebra I – MAC1105 Homework Instructions: Answer the questions below and show your work when calculations are required. Try to write your answer on the blank space below each question. 1. You have two options for investing $500. The first earns 7% interest compounded annually, and the second earns 7% simple interest. The figure shows the growth of each investment over a 30year period. a. What is the approximate amount of money in your account, after 15 years, for the linear model? Explain b. What is the approximate amount of money in your account, after 20 years, for the exponential model? Explain c. How much more money is, approximately, in your account for the exponential model than for the linear model, after 30 years? How did you calculate this amount? 1 2. According to the U.S. Census Bureau, the projected world population t years after 2010 is given by the function: 𝒇(𝒕) = 𝟔. 𝟗𝟒𝟕𝒆𝟎.𝟎𝟎𝟕𝟒𝟓𝒕 a. Is this equation representing a population growth or a population decay? Explain based on the sign of the constant k = 0.00745. b. What value of t will you use to calculate the population in 2025? Remember t is the number of years after 2010. 3. You will deposit $500 in a bank account at a 7% interest compounded annually. The formula for compound interest is shown below: 𝑟 𝑛𝑡 𝐴(𝑡) = 𝑃 (1 ) 𝑛 a. Select the values of P, r, and n from the problem statement. b. If you graph t vs. A, will the graph be increasing, decreasing or constant? Explain. c. Write the formula that describe the data given. 2 4. Suppose 600g of a radioactive substance are present initially and 3 years later only 300g remains. If the equation modelling the change in mass of the radioactive substance is 𝑦 = 𝑦0 𝑒 𝑘𝑡 : a. What does y0 represent, and what is its value, based on the problem’s statement (above)? b. Is this problem about exponential growth or exponential decay? Explain. c. What does k represent and what is going to be its sign (positive or negative)? Explain. 5. You will deposit $3000 in a bank account at a 2% interest compounded monthly. The formula for compound interest is shown below: 𝑟 𝐴(𝑡) = 𝑃(1 )𝑛𝑡 𝑛 a. Select the value of P, r and n from the problem statement. b. Explain the order in which you will perform the operations to calculate A(t). c. What will be the amount in the bank account after 4 years? 3 6. Suppose 600g of a radioactive substance are present initially and 3 years later only 300g remains. If the equation modelling the change in mass of the radioactive substance is 𝑦 = 𝑦0 𝑒 𝑘𝑡 : a. Explain the steps to determine the value of the constant k, based on the information provided. b. How much of the substance will be present after 6 years. 7. You have two options for investing $500. The first earns 7% interest compounded annually, and the second earns 7% simple interest. The figure shows the growth of each investment over a 30year period. a. Determine which graph represents each type of investment. Explain your reasoning. b. Which option would you choose? Explain c. Just based on the information provided by the graph (without any calculation), can you estimate the amount of money in your 7% interest compounded annually account after 45 years? Explain. Final Investment Value using Simple Interest formula: 𝐴(𝑡) = 𝑃(1 𝑟𝑡) 𝑟 𝑛𝑡 Final Investment Value using Compounded Interest formula: 𝐴(𝑡) = 𝑃 (1 𝑛) 4 8. According to the U.S. Census Bureau, the projected world population t years after 2010 is given by the function: 𝒇(𝒕) = 𝟔. 𝟗𝟒𝟕𝒆𝟎.𝟎𝟎𝟕𝟒𝟓𝒕 a. Could you use the given function to calculate what the world population was in 2005? Explain. 9. The exponential growth model for a population contains certain assumptions, for example: 1. There is constant rate of birth and death among the individuals of the population. 2. There is no immigration and emigration taking place in the population. 3. There are no genetic variations among the individuals of the population. 4. Variations in age and size among population members are not included a. Explain how failing to make these assumptions will affect your exponential growth model (use at least two of the assumptions above). The formula for the exponential growth model is: 𝐴 = 𝐴0 𝑒 𝑘𝑡 , where: 𝐴0 – original size. k – constant representing the growth rate. t – amount of time 5 10. From 1950 to 2010, the world population approximately increased by 165%. The data presented below supports the above statement. The data is presented in three different forms (table, set of ordered pairs, and a scatter plot graph). a. Is the statement above correct, based on the data provided? Explain. b. What format(s) could you use in a presentation to support the above statement? Explain your selection. c. What format(s) let you identify better the trendline that better fit the data (linear, exponential, logarithmic, polynomial, etc.)? Explain your selection. a. Table: x, year 1950 1960 1970 1980 1990 2000 2010 y, world population (billions) 2.6 3.0 3.7 4.5 5.3 6.1 6.9 b. Set of ordered pairs: {(1950, 2.6), (1960, 3.0), (1970, 3.7), (1980, 4.5), (1990, 5.3), (2000, 6.1), (2010, 6.9)} (x, y) = (year, population in billions). c. Graph: 6 7