Algebra 2A

Exponential and Logarithmic Functions End of Unit Project

Exponential Growth:

You have $5,000 to invest at your local bank. Your bank offers you two different investment options. For the first option, your bank offers an annual interest rate of 3% compounded monthly. For the second option, the bank offers a simple annual interest rate of 3.5%.

1) Write an equation for each scenario using the following formulas:

Compound Interest:

Simple Interest: A = P + (Prt)

P = principal amount (the initial amount you deposit)

r  = annual rate of interest (as a decimal)

t  = time(in years)

A = amount of money in the account after t years, including interest.

n  =  number of times the interest is compounded per year (daily = 365, weekly = 52, monthly = 12, quarterly = 4, semiannually = 2, annually = 1)

Compound Interest Equation: A=P(1+r/n0^(nt) (5 points)

Simple Interest Equation: A=P+(prt) (5 points)

2) Which equation is linear (2 points)? Which Equation is exponential (2 points)?

· Compound Interest in Linear, Simple Interest is exponential.

3) Make a prediction about which account you think will grow your money the fastest. Explain your reasoning (6 points).

· I think the Compound Interest will grow money faster because it has a less interest rate so you save more money.

4) Make a table of values that shows the amount of money you will have in your account each year for the next 5 years for each account (make sure to include year 0, which is the amount of money you start with). How much money will you have in each account after 20 years? (14 points)

5) Make one graph with both sets of data. Make sure to label each graph and include all important components of a graph (label the axes, mark the scale on the axes, title, etc.) (10 points).

6) Which account would you choose to put your money in and why? (6 points)

Exponential Decay:

Materials: Bouncy ball, yard stick or tape measure

You will also need another person to help you complete this project.

Can you think of sports where the ‘bounciness’ of the ball is an important factor in the game?  Did you know that in many sports, there are official rules about the bounciness of balls for regulation play? Have you ever watched a ball bounce repeatedly?  Due to the loss of energy each time a bouncing ball hits the floor, the ball never rebounds to the same height from which it fell.

The ‘bounciness’ or ‘bounce factor’ of a ball can be determined by comparing the ball’s rebound height to the original height from which the ball was dropped.  Then, using the bounce factor of the ball, you can predict the height of the ball after any number of bounces.

Conducting the Experiment:

1. Attach the tape measure or meter sticks to the wall

2. Drop the ball from a height above your head (record on the handout). Measure how high the ball bounces and record on the handout. Now hold the ball at the height you just recoded and drop the ball again. Record the bounce height. Continue with this process until the ball is too low to measure the height accurately.

Analyzing data:

1. Calculate the bounce factor for each drop of the ball (on table).

Type of ball we had: ______________

Data Collection (20 points):

2. Mean Bounce Factor (b) = _________ (5 points)

3. Mathematical Model: Use the form y = abx where a is the initial drop height and b is the mean bounce factor. (10 points)

4. Make two graphs on the same set of axes. One graph is your experimental data (column 3) and one graph is your calculated data (column 5). Use the bounce number for x and the bounce height as y (you should NOT have a straight line). (10 points)

5. Was there a difference between your experimental data and your calculated data? Why do you think this is? (5 points)

For questions that require calculations, all calculations should be shown, not just the final answer. For questions that require an answer using SPSS, the Variable View, Data View, and Output Panel screens should be copied and pasted to your exam.

Answer all questions on the exam thoroughly. Create a Microsoft Word document, including the question number, the question, your typed answer, and SPSS screens if required. You may use Word’s equation editor to complete your answers.

1) A newspaper article about the results of a poll states: "In theory, the results of such a poll, in 99 cases out of 100 sho uld differ by no more than 5 percentage points in either direction from what would have been obtained by interviewing all voters in the United States." Find the sample size suggested by this statement.

2) A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $664.14 and a standard deviation of $297.29. Using SPSS, find a 98% confidence interval for the true mean checking account balance for local customers.

3) Test the claim that for the adult population of one town, the mean annual salary is given by Sample data are summarized as and Use a significance level of

4) Use the P-value method to test the claim that the population standard deviation of the systolic blood pressures of

adults aged 40-50 is equal to The sample statistics are as follows: Be sure to state the hypotheses, the value of this test statistic, the P-value, and your conclusion. Use a significance level of

0.05.

5) The maximum acceptable level of a certain toxic chemical in vegetables has been set at 0.4 parts per million (ppm). A consumer health group measured the level of the chemical in a random sample of tomatoes obtained from one producer. The levels, in ppm, are shown below.

0.31 0.47 0.19 0.72 0.56

0.91 0.29 0.83 0.49 0.28

0.31 0.46 0.25 0.34 0.17

0.58 0.19 0.26 0.47 0.81

Do the data provide sufficient evidence to support the claim that the mean level of the chemical in tomatoes from this producer is greater than the recommended level of 0.4 ppm? Use a 0.05 significance level to test the claim that these sample levels come from a population with a mean greater than 0.4 ppm. Use the P-value method of testing hypotheses. Assume that the standard deviation of levels of the chemical in all such tomatoes is 0.21 ppm.

6) A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. Construct a 99% confidence interval for the difference between the two population proportions.

7) The confidence interval, 3.56 < σ < 5.16, for the population standard deviation is based on the following sample

statistics: n = 41, = 30.8, and s = 4.2. What is the degree of confidence?

8) A random sample of 16 women resulted in blood pressure levels with a standard deviation of A random sample of 17 men resulted in blood pressure levels with a standard deviation of Using SPSS and a 0.05 significance level, test the claim that blood pressure levels for women vary more than blood pressure levels for men.

9) The following data shows annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Assume that incomes are not affected by an interaction between gender and level of education, and test the null hypothesis that gender has no effect on income. Use a 0.05 significance level.

10) At the 0.025 significance level, use SPSS to test the claim that the four brands have the same mean if the following sample results have been obtained.

�

�

�

�

�

�

�

�