Download 3B- Ch 8-4 APPS exponential functions

Math 3B
Problems involving exponential functions
Name
11/_____/14
The general exponential function: y = ab x ( b will always be positive.)
b is the base. When the exponent involves time a is often called the initial value,
because when x = 0, then y = a.
€ Formula.
Yearly Compound Interest/Population Growth
x
y = P (1+ r)
For money: y is the amount of money after x years resulting from an investment of P at a yearly
interest rate of r. Note r is a percent like 5% = 0.05.
For population: y is the population after x years resulting from an initial population of P
at a yearly growth rate of r.
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Problems
1. The population of Vancouver was 397,000 in 1987.
Assume the population grew exponentially at the rate of 1.6% per year from that date.
a. Write an equation predicting the population P(t), after t years, where t = 0 stands for 1987.
b. What was the population in 1997?
c. Complete this table, you may use the table feature of your calculator.
t
P(t)
Δ
Ratio
XXXX
XXXX
XXXX
XXXX
0
1
2
3
9
10
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?
f. At what rate per year does the population grow?
2. The population of a city is now 250,000 and is decreasing at a rate of 5% per year exponentially.
This can be dealt with just like population growth. You use a negative r value.
a. Write an equation describing the population over time. Use function notation.
b. Is the base for this equation larger or less than 1? So, will the equation of the graph rise or fall
with time?
c. What will the population be in 30 years?
d Graph to find when the population will be 50,000 if the rate of decrease continues at 5%.
3. Write an exponential function that passes through the two points (0, 5) and (3, 18).
4. A sample of bacteria increases in number exponentially. Over a 30 hour period an initial population
of 800 grows to 2400. Use the same method you used in #3 above.
a. Write an exponential function which predicts the number of bacteria B(t) after t hours.
b. How many are there after 15 hours?
c. How long did it take for the population to double to 1600?
(Graph and find the intersection of your curve with y = 1600.)
5. A bank pays compound interest at a rate of 2.9% per year. You deposit $4000.
a. Write an exponential function to predict your account balance.
b. How much will you have after 5 years?
c. When will you have $8000?
6. How much to invest?
How much should be invested at a rate of return of 7% to have $100,000 after 15 years?
Hint: The unknown is the initial investment P.
7. What rate of return is needed?
If you want to invest $1000 and have $2000 in 10 years what yearly rate of return is needed?
Hint: If you write the necessary equation it is r you must solve for.
Opener Math 3B/4A
Name:__________________________________________
Solve algebraically – Not by Graphing.