NAME__________________________________DATE_____________PERIOD_____
7 -5 C Applications of Log and Exponential Functions
Use the formula for compound interest
r

A  P 1  
n

CC12
nt
to solve. Show the (a) formula with
numbers substituted, and (b) the answer. Round to the nearest cent.
1. $500 is invested at 6% annual interest, compounded quarterly.
a. What is the balance after one year?
b. What is the balance after seven years?
c. When will the balance be $1000?
2. How much money must be deposited in an account that pays 9% annual interest, compounded
monthly, to have a balance of $3000 after three years?
3. If $1000 is inivested in an account that pays 3.5% interest, how much will there be after 10 years
if it is
a. compounded monthly?
b. compounded quarterly?
4. The value of a new $22,500 automobile decreases 20% per year.
a. Find its value after one year.
b. Find its value after 8 years.
c. When will it be worth only $2000 ?
5. A sailboat depreciates at the rate of 12% per year.
a. What would the value of a new $35,000 sailboat be after two years?
b. When will it be worth $5000 ?
6. A certain bacteria population doubles in size every 12 hours. The initial population is 6 bacteria.
a. How many bacteria will there be after 24 hours?
b. How many bacteria will there be after three days?
c. When will the bacteria population reach 1,000,000?
Use the half-life formula
 1
N  No  
2
t
h
where
No
is the initial amount, h is the half-life, t is
the time and N is the final amount. Round to 3 decimal places.
7. Cobalt-60 has a half-life of about 5 years. How much of an 8-gram sample will remain after 30
years?
8. One isotope of chromium has a half-life of 20 hours. How much of a 100-gram sample is left after
100 hours?
9. How long would it take 80 grams of the chromium (half-life of 20 hours) to decay to 50 grams?
10. A fossil contains 47 mg of carbon-14. Determine the age of the fossil if it originally contained 93
mg of carbon-14 and the half-life of carbon-14 is 5570 years.
11. A fossil found at an archeological dig has 74 mg of carbon-14. If it originally contained 150 mg,
use the carbon-14 dating formula to determine its age.
The pH of a patient’s blood can be calculated using the Henderson-Hasselbach Formula:
pH = 6.1 + log
B
C
where B is the concentration of bicarbonate and C is the concentration of carbonic acid. The
normal pH is approximately 7.4.
10. A patient has a bicarbonate concentration of 24 and a carbonic concentration of 1.9. Find the pH
of the patient’s blood.
11. A patient with a normal blood pH has a bicarbonate concentration of 24. Find the concentration
of carbonic acid.
When it is compounded continuously, the amount in an account after t years is
where r is the annual interest rate and P is the principal.
A = Pert,
12. You deposit $1200 in an account that pays 5% interest. Find the amount in the account after 10
years if it is compounded continuously.
13. You deposit $2000 in an account that pays 3.5% interest. Find the amount in the account after 5
years, if it is compounded continuously.
14. If money is invested in an account paying 4.5% interest compounded continuously, how long will
it take for the investment to double?
15. The population of bacteria can be represented by the formula N = Noekt, where No is the initial
number of bacteria in the culture. N is the number after t hours, and k is a constant determined
by the type of bacteria and the conditions. When will a culture of 300 bacteria, where k = 0.068,
reach a count of 10,000?
16. A college math class consists of 32 students. On Monday at 9 AM, the teacher tells one student
to notify the others that the test scheduled for Wednesday at 9 AM has been cancelled. The
model for the number of students in the class who have heard this information after t hours is
N = 32 – 32e-0.02t. After how many hours will half of the class have been notified?
17. A container of ice cream arrives home from the supermarket at a temperature of 65 0F. It is
placed in the freezer which has a temperature of 200F. How long will it take the ice cream to
reach a temperature of 320, if the rate of cooling is 0.1070F per minute?
Use Newton’s law of cooling: Tf = Tr + (To – Tr)e-rt.
18. The power of supply of a satellite decreases exponentially over the time it is being used. The
equation for determining the power supply P, in watts, after t days is P =
the number of days it will take for the power supply to be less than 30W.
50e

t
250
. Determine
19. The cooling model for tea served in a 6 oz cup is Tf = Tr + (To – Tr)e-.41t. If the original
temperature of the tea was 2000F and the room temperature is 680F, determine after how many
minutes the tea will be 850F.
20. The internal temperature of a roast beef is 1500F when it is removed from the oven. It is set on a
kitchen counter where the room temperature is 770F. Using Newton’s law of cooling,
Tf = Tr + (To – Tr)e-rt with a cooling rate of 0.06, determine after how many minutes the
temperature of the roast beef will be 2/3 of the original temperature.