Ma 093 and MA 117A - Exponential Models Topic 1 – Compound Interest
1) Compound Interest – A person invests $7000 at 10% interest compounded annually. a) Find an equation for the value of the investment after t years.
*
(1
)
7000(1 .10)
7000(1.1)
x t t ty
a b
A
P
r
A
A
=
=
+
=
+
=
t = years after investment was made A = amount of money in account Coordinates of points (t years, A dollars)
b) Show graph and label.
c) What is the y-intercept? What does it mean within context?
If t = 0, then A = 7000. The y-intercept is 7000. The point is (0 years, $7000) Meaning: The initial deposit into the account was $7000
d) What is the meaning of b in the equation * x
y=a b ?
In this problem, b = 1.1
Interpret as a rate: 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.
Interpret as a multiplier: The amount in the account in any year is obtained by multiplying the amount of the previous year by 1.1
e) How much will be in the account in 8 years?
8
7000(1.1) $15, 005.12
A = =
In 8 years the account will have $15,005.12
f) When will the amount in the account double? 14000=7000(1.1)t Solve with the calculator –
GRAPHICAL APPROACH Enter Y1 = 7000(1.1)t
Y2 = 14000
Press WINDOW and use [0, 15] for x and [0, 15000] for y
Press GRAPH
It will take about 7.3 years to double
Now solve using algebra: (isolate exponential and take logarithms) 2 1.1 1.1 ) log 2 * log1.1 log 2 log1.1 7.3 log 2 log( t t t t t = = = = =
2) Compound Interest – A person invests $..10,000... at ...5..% interest compounded annually.
Find an equation for the value of the investment after t years.
* (1 ) 10000(1 .05) 10000(1.05) x t t t y a b A P r A A = = + = + =
t = years after investment was made A = amount of money in account Coordinates of points (t years, A dollars)
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 y=a b* x ?
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? 10
10000(1.05) $16, 288.95
A = =
In 10 years the account will have $16,288.95
b) TYPE: Given A, find t: When will the amount have $13,000 13000 10000(1.05)= t?
Solve with the calculator – GRAPHICAL APPROACH Enter Y1 = 10000(1.05)t
Y2 = 13000
Press WINDOW and use [0, 15] for x and [0, 15000] for y Press GRAPH
3) 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 A=P(1+r)t t = 18 10000(1 .08) 10000(1.08) t t A = + = 18 10000(1.08) A = = $39,960.19
Topic 2 – Constructing an exponential function
4) The population of a country is growing exponentially. In the year 2000 there were 3.5 million people and the population doubles every five years. Produce an equation to model the
population of this country in terms of years since 2000.
* x
y=a b
Years since 2000 Population (millions of people)
0 3.5
5 7 (it doubles from 2.5)
10 14
3.5*
xy
=
b
5 5 1/ 57
3.5*
3.5
2
2
b
divide by
b
b
=
=
=
b = 1.149 3.5*1.149x y =Put this in the Y= of the calculator and look at the TABLE to see if it makes sense, compare the y-values for x = 0 and x = 5 with the ones on the table above
`
The calculator table shows the point (5, 7.0092) while the table on the top shows (5, 7). Why is the y value different?
5) The number of bacteria in is growing exponentially. Assume that now there are 2000 bacteria and the number triple every hour. Write an equation to model the number of bacteria in the dish in terms of the number of hours from now.
X hours from now Y = number of bacteria
0 2000 1 6000 (it triples)
*
2000*
x xy
a b
y
b
=
=
Since the bacteria triples EVERY hour, then, the multiplier is 3 The equation is:
2000*3
x
y =
6) The number of bacteria in is growing exponentially. Assume that now there are 2000 bacteria and the number triple every 10 hours. Write an equation to model the number of bacteria in the dish in terms of the number of hours from now.
X hours from now Y = number of bacteria
0 2000 10 6000 (triples)
*
2000*
x xy
a b
y
b
=
=
10
6000
=
2000*b
103
=
b
1/10
3
=
b
b = 1.116
2000*1.116
x
y =
Exponential Decay – Half Life
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:
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 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 80 120
Y 200 100 50 25
d) 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
e) 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
f) Use the exponential regression feature of the calculator to find the exponential function y = a b• that x fits the data. Use the points from part (c) to enter in the Editor of the calculator. Round the constants to 3 decimal places.
200*.983
x
y =
Let’s continue with this!!!! WHAT FOLLOWS IS VERY IMPORTANT!!!! INTERPRET a and b WITHIN CONTEXT
200*.983
x
y =
Today, there are 200 grams of radioactive substance in the tank and every year the amount remaining is multiplied by 0.983.
Notice that 0.983 is the same as 98.3%
Every year, 98.3% of the radioactive substance remains and 1.7% decays (100% - 98.3% = 1.7%)
Let’s use the function to check our estimates to (d) and (e) on the previous page: d) Estimate the amount remaining 70 years from today. Use proper units.
70
200*.983
60.2
y
=
=
grams
a) Estimate when the amount remaining will be 20 grams? Use proper units.
20
=
200*.983
x
Put in your calculator Y1 = 200 * 0.983^x Y2 = 20
And use the INTERSECT feature of the calculator to find the intersection
8) 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 remaining 25 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 be 40 grams? Use proper units
If y = 40, x ~ 12
In 12 years there will be about 40 grams left
If y = 40, x ~ 60
f) Use algebra to produce the equation for each of these exponential models.
X years from now Y = amount remaining in grams
0 100 10 50
