MA 093 and Math 117A – Section 4.5 – Exponential Models - Compound Interest & Exponential Decay
Solutions
- Compound Interest – A person invests $7000 at 10% interest compounded annually.
Find an equation for the value of the investment after t years.
WHAT ELSE CAN I ASK ABOUT THIS PROBLEM?
(1) Graph an exponential increasing function, label axes and the y-intercept.
(2) What is the y-intercept? What does it mean within context?
The y-intercept is 7000.
Meaning: The initial deposit into the account was $7000
(3) What is the meaning of b in the equation ?
In this problem, b = 1.1
The amount of money in the account increases at a rate of 10% per year.
1.1 = 1+ 0.1 = 1 + r. Since r = 0.1, then the rate is 10% per year
(4) Think on questions of different types:
a) TYPE: Given t, find A: How much will be in the account in 8 years?
In 8 years the account will have $15,005.12
b) TYPE: Given A, find t: When will the amount in the account double?
Solve with the calculator – GRAPHICAL APPROACH
Enter Y1 =
Y2 = 14000
Press WINDOW and use [0, 15] for x and [0, 15000] for y
Press GRAPH
It will take about 7.3 years for the amount in the account to double
- Compound Interest – A person invests $10,000 at 5% interest compounded annually.
Find an equation for the value of the investment after t years.
WHAT ELSE CAN I ASK ABOUT THIS PROBLEM?
(1) Graph an exponential increasing function, label axes and the y-intercept.
(2) What is the y-intercept? What does it mean within context?
The y-intercept is 10,000.
Meaning: Initially, we deposited $10000 into the account
(3) What is the meaning of b in the equation ?
In this problem, b = 1.05
The amount of money in the account increases at a rate of 5% per year.
1.05 = 1+ 0.05 = 1 + r. Since r = 0.05, then the rate is 5% per year
(4) Think on questions of different types:
a) TYPE: Given t, find A: How much will be in the account in 10 years?
In 10 years the account will have $16,288.95
b) TYPE: Given A, find t: When will the amount have $13,000
?
Solve with the calculator – GRAPHICAL APPROACH
Enter Y1 =
Y2 = 13000
Press WINDOW and use [0, 15] for x and [0, 15000] for y
Press GRAPH
In about5.4 years the account will have $13,000
- Compound Interest – on the day you were born, your grandparents set a college fund for you. They deposited $10,000 in an account that paid 8% compounded annually. How much will you have available for college when you turn 18?
P = 10,000
r = 0.08
t = 18
= $39,960.19
- The population of a country was 2.5 million people in the year 2000 and since then it has been increasing at a rate of 2% annually. That is, each year, the population is about ______times the previous year’s population. Write an equation for the population of the country (in millions) at t years since the year 2000.
WHAT ELSE CAN I ASK ABOUT THIS PROBLEM?
(1)When will the population double? Use the graphical approach
- Write equation:5 = 2.5(1.02)^x
- Enter y1 = 5
- Enter y2 = 2.5(1.02)^x
- Use window: [0,50,1} for x and [0,10,1] for y
- Find the intersection point x = 35.
- Answer: If the same model continues, the population will double in the year 2035. Extrapolation, not very confident in this estimate.
- Sketch the graph and label in context
- The revenue from music downloads was $1.98 billion in 2007 and has grown by about 86% per year since then. That is, each year the revenue is about ______times the previous year’s revenue.
a)Find an equation for the revenue, R(t) (in billions of dollars) in the year that is t years since 2007.
b)Find R(7) and interpret in context with correct units.
- A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after today. The graph for f is shown below:
a)How many grams of the radioactive substance does the tank contain today? Use proper units.
The y intercept is 200. Today there are 200 grams of radioactive substance in the tank.
b)Use the graph to determine the half life of the element? Use proper units.
It takes 40 years for the 200 grams to decay to half (to 100 grams). The half life is 40 years
c)Use the graph to estimate the amount remaining 70 years from today. Use proper units.
Estimate the y value when x is 70. In 70 years there will be about 60 grams in the tank
d)Use the graph to estimate when the amount remaining will be 20 grams? Use proper units.
Estimate the x value when y = 20. It will take about 130 years for the tank to contain 20 grams
e)Use the graph to read the coordinates of 4 points related to the half-life information starting with the
Y-intercept.Record the coordinates on the table.
X / 0 / 40 / 80 / 120Y / 200 / 100 / 50 / 25
f)Use algebra and the first two points from the table in part (e) to find the exponential function
y = that fits the data; round to three decimal places.
g)Complete the following: Today, there are ____200 grams__ of radioactive substance in the tank. Every year, the amount remaining is decreasing by subtracting/multiplying (circle one) by __1/2____. Every year_ __98.3% of the substance remains and ___1.7% decays. (100% - 98.3% = 1.7%)
h)Use the exponential regression feature of the calculator to find the exponential function y = that fits the data. Use the points from part (c) to enter in the Editor of the calculator. Round the constants to 3 decimal places.
Let’s use the function to check our estimates to (c) and (d) on the previous page:
i)Use the model equation to find the amount remaining 70 years from today. Compare the answer to your estimate from part (c).
j)Use the model equation and the graphical approach to determine when there will be 20 grams left. Compare the answer to your estimate from part (d).
Graphical Approach
- Write the equation
- Enter the left hand side of the equation in Y1 of the calculator
- Enter the right hand side of the equation in the Y2 of the calculator
- Enter appropriate window values.
- Press 2nd TRACE [CALC]
- Select 5:intersect
- Press ENTER three times until you find the point of intersection
- Answer the problem
Put in your calculator Y1 = 200 * 0.983^xY2 = 20
And use the INTERSECT feature of the calculator to find the intersection
It takes about 134.3 years to decay to 20 grams
7. A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after today. The graph for f is shown below.
(1) Use x-scale = y-scale = 10 and label the tic-marks along the axes. Each tic mark has a value of 10/ (2) Use x-scale = 20, y-scale = 30 and label the tic-marks along the axes. Use the scales to label the tic-marks on the graph.
a)How many grams of the radioactive substance does the tank contain today? Use proper units.
The y-intercept will be 100
Today, the tank has 100 grams of radioactive substance. / The y-intercept will be 300. There are 300 grams of radioactive substance in the tank.
b)Use the graph to determine the half life of the element? Use proper units.
When x = 10, y = 50 (50 is ½ of 100)
The half-life is 10 years / When x = 20, y = 150 (150 is ½ of 300)
The half-life is 20 years
c)Use the graph to read the coordinates of 4 points related to the half-life information starting with the Y-intercept.Record the coordinates on the table.
x / 0 / 10 / 20 / 30
y / 100 / 50 / 25 / 12.5
/ x / 0 / 20 / 40 / 60
y / 300 / 150 / 75 / 37.5
d) Use the graph to estimate the amount remaining25 years from today. Use proper units.
If x = 25, y ~15
In 25 years there will be about 15 grams remaining. / If x = 25, y~120
In 25 years there will be about 120 grams remaining
e) Use the graph to estimate when the amount remaining will be40 grams? Use proper units
If y = 40, x ~ 12
In 12 years there will be about 40 grams left / If y = 40, x ~ 60
In 60 years there will be about 40 grams left
Problem number 7 continued:
f) Use algebra and the first two points from the table in part (c) to find the exponential function that fits the data; round to three decimal places.
X years from now / Y = amount remaining in grams
0 / 100
10 / 50
/ X years from now / Y = amount remaining in grams
0 / 300
20 / 150
g) Use the points from part (c) and the exponential regression feature of the calculator to find the exponential function y = that fits the data; round to 3 decimal places.
Enter data, and do ExpReg L1, L2, Y1
You should get the same equation from part (f) / Enter data, and do ExpReg L1, L2, Y1
You should get the same equation from part (f)
h) Interpret the constants a and b from the exponential model following the format shown on part (g) of problem (6)
Today, there are 100 grams of radioactive substance in the tank. Every year the amount decays by multiplying by 0.933.
Notice that 0.933 = 93.3%
Every year, 93.3% of the radioactive substance remains and 6.7% decays.
(100% - 93.3% = 6.7%) / You do this one
Problem 7 continued
i)Use the model equation to find the amount remaining 25 years from today. Use proper units.
You do this one / You do this one
d)Use the model equation and the graphical approach to determine when there will be 40 grams in the tank.
Graphical Approach
- Write the equation
- Enter the left hand side of the equation in Y1 of the calculator
- Enter the right hand side of the equation in the Y2 of the calculator
- Enter appropriate window values.
- Press 2nd TRACE [CALC]
- Select 5:intersect
- Press ENTER three times until you find the point of intersection
- Answer the problem
40 = 100 * (.933)^x
Enter LHS in Y1
Enter RHS in Y2
Use convenient window values
Find point of intersection / Graphical Approach
- Write the equation
- Enter the left hand side of the equation in Y1 of the calculator
- Enter the right hand side of the equation in the Y2 of the calculator
- Enter appropriate window values.
- Press 2nd TRACE [CALC]
- Select 5:intersect
- Press ENTER three times until you find the point of intersection
- Answer the problem
40 = 300 (.966)^x