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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 predict
ion 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)
|
Years (t) |
Amount in Compound Interest Account |
Amount in Simple Interest Account |
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:
- Attach the tape measure or meter sticks to the wall
- 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:
- Calculate the bounce factor for each drop of the ball (on table).
Type of ball we had: ______________
Data Collection (20 points):
|
Bounce Number (x) |
Drop Height |
Bounce Height (y) |
Bounce factor = bounce height/drop height |
Calculated Bounce Height using formula from #3 below (y) |
|
(a) |
||||
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)
Algebra
2
A
Exponential and Logarithmic Functions End of Unit Project
Exponential Growth:
You have $
5
,
0
00 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 = 3
6
5, 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)
|
Years (t) |
Amount in Compound Interest Account |
Amount in Simple Interest Account |
| 0 | ||
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 20 |
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):
|
Bounce Number (x) |
Drop Height |
Bounce Height (y) |
Bounce factor = bounce height/drop height |
Calculated Bounce Height using formula from #3 below (y) |
|
(a) |
||||
| 6 | ||||
|
7 |
||||
|
8 |
||||
|
9 |
||||
| 10 |
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)
