c. Graph this on your calculator and determine about when the average was 600 pages.

Size: px

Start display at page:

Download "c. Graph this on your calculator and determine about when the average was 600 pages."

Allison Ramsey
5 years ago
Views:

1 EXPONENTIAL MODELING: CLASS PROBLEMS 1. In 1950 the average Algebra II book had 412 pages. The current Algebra II book has 850 pages. a. What was the annual percentage growth in the number of pages? b. Write an equation describing the number of pages the average book had P(t) as a function of the number of years since 1950 (t). Assume that the annual growth rate has been constant. c. Graph this on your calculator and determine about when the average was 600 pages. d. Find when the average was 600 pages in a different way. Explain your method. 2. What rate of return is needed? If you want to invest $1000 and have $2000 in 9 years what yearly rate of return is needed?

2 3. Write an exponential function that passes through the two points (0, 4) and (7, 500). 4. How much should be invested at 8% per year return to have $40,000 after 15 years?

3 Exponential Modeling Problems: HOMEWORK Directions: This handout covers all the types of exponential modeling that we ve talked about in class. 1. Depreciation. Some brands of new cars lose 20% of their value each year. After one year they are only worth 80% of their original price and the next year 80% of that, etc. It is said that a car "depreciates in value 20% per year." a. Write an equation expressing the value y of a $45,000 car after x years whose value depreciates 20% per year. b. What will the car be worth in 10 years? 2. The population of Vancouver was 397,000 in Assume the population grew exponentially at the rate of 1.6% per year from that date. a. Write an equation predicting the population y, after x years, where x=0 stands for b. What was the population in 1997? d. About how much did the population grow (change) in the first year, in persons per year? e. About how much did the population grow (change) in the one year period before 1997?

4 3. Suppose that as a New Year s Resolution you decided to save your money in the following way. Before the start of the year, you had $0.25 in a piggy bank. Each day you would double the money in your piggy bank. So on January 1, you would have $0.50 saved. On January 2, you would have $1.00 saved, and so on. Let y stand for the amount you would have saved as of day x of January. a. Without a calculator, fill in the output column of the table b. Write a function formula for y. y = c. Graph the points on the grid. d. Sketch the graph of y by drawing a curve that goes through the points. e. If you kept up this process, how much money would you have on January 31? Date x y Dec Jan Jan Jan 3 3 Jan 4 4 Jan 5 5 Jan 6 6 Jan How many years will it take to reach an investment goal depends on the interest rate. a. Suppose you invest $15,000 at 4% per year. When will you double your money? b. Suppose you invest the same $15,000 at 8% per year. Now how long until you double your money? c. What if you want to turn $1000 into $1 million in ten years? What interest rate is required? Be sure you read your result correctly.

5 5. Did the Dutch pay too much or too little for Manhattan island? This is a classic question to illustrate the earning power of money at various rates of return. It s a bit unrealistic for many reasons. It is recorded that Manhattan Island was purchased from a tribe of Native Americans for jewelry and tools worth about $24. Considering the value of Manhattan s land today was this a bargain or not? One way to answer the question is to ask what the Native Americans would have now if they had invested their $24 in a bank account for the last 386 years (since 1626). Then compare that to the current value of all the land on Manhattan, which is around $8 trillion. If it is more than this amount the Dutch paid too much and if less they got off paying too little. The answer depends entirely on the interest rate the natives could have had during all those years. a. Suppose they were able to invest at 3%. What would their $24 have grown to? b. How about 8%? (Amazing difference, no?) c. What rate would give them exactly $8 trillion so that the $24 was just the right price

6 6. High rates of exponential growth cannot continue for long. According to the Boston Globe anthrax bacteria is capable of doubling every 15 minutes in the human body. This rate cannot keep up for many hours as you can calculate. a. Write an equation which predicts the number of bacteria y after x hours if the starting number is 500. b. How many would there be after 4 hours? c. After 24 hours? Could this happen? (Note there are several trillion cells in your body and a trillion is ) 7. The value of a painting by a famous artist increased from $20,000 to $40,000 in 7 years. a. What was the annual rate of increase in the painting s value? b. Write an exponential function showing its value when it is t years old. (Assume the annual rate of increase stays constant). c. What will its value be in 10 years?

7 8. According to Scientific American, there were between 5 and 10 million wild elephants in Africa in By 1990, when they were added to the international list of critically endangered species, the African elephant population was thought to be about 600,000. Estimate the average percentage decline in the elephant population during that time, and use your model to predict the number of elephants that will remain in Africa in 2030.

Survey of Math Chapter 21: Savings Models Handout Page 1

Survey of Math Chapter 21: Savings Models Handout Page 1 Chapter 21: Savings Models Handout Page 1 Growth of Savings: Simple Interest Simple interest pays interest only on the principal, not on any interest which has accumulated. Simple interest is rarely used