PreAP Precal Application of Logs
Growth and Decay
PLEASE SHOW WORK NEATLY ON A SEPARATE PAPER BOX ANSWERS
1. A radioactive substance has a half-life of 420 years. How much remains of a 2 oz.
sample after 200 years?
2. A bacteria culture has an initial count estimate of 4000. After 20 minutes the count is
22, 400. What is the growth rate, and approximately how many minutes did it take for the
culture to double?
3. An isotope of sodium has a half-life of 15 hours. How many hours will it take for 40% of
a given amount to remain?
4. The number of bacteria in a certain culture increases from 600 to 1800 in 2 hours. How
many bacteria will there be after 4 hours?
5. Radium has a half-life of 1600 years. If you begin with 50 mg, when will there be only
20 mg left?
6. The number of bacteria increases form 5,000 to 15,000 in 10 hours.
a) What is the bacteria’s growth rate?
b) How many bacteria will there be in 20 hours?
c) When will there be 50,000 bacteria?
7. The polonium isotope 210Po has a half-life of 140 days. If the initial sample is 20 mg, how
much will be left after two weeks?
8. The population of a city increases 5% per year. If the present population is 500,000,
what will the population be in 10 years?
9. If ¼ of a radioactive substance disintegrates in 10 days, what is its half-life?
10. A physicist has 12 grams of radioactive bismuth, which has a half-life of 5 days.
a) How much will be left after 15 days?
b) How long will it take for 2 grams to disappear?
11. In 1950 the population of a city was 80,000, and in 1960 it was 100,000.
a) What is its yearly growth rate?
b) What is the population in 1980?
c) What is the population in 2010?
Log Applications – Physics and Miscellaneous
12. Oliver Sudden is driving along a straight, level highway at 64 km/hr when his car runs
out of gas. As he slows down, his speed decreases exponentially with the number of
seconds since he ran out of gas, dropping to 48 km/hr after 10 seconds.
a) Write the particular equation expressing speed in terms of time.
b) Predict Oliver’s speed after 25 seconds.
c) At what time will Oliver’s speed be 10 km/hr?
d) Draw the graph of the function for speed in the domain from 0 through the time
when Oliver reaches 10 km/hr.
13. Phoebe Small is out Sunday driving in her rocket ship. She fills up with fuel at the
Scorpion Gulch Rocket Fuel Station, and takes off. When she starts the last stage of her
rocket, she is going 4,230 mph. Ten seconds later she is going 6, 850 mph. While the last
stage is running, you may assume that her speed increases exponentially with time.
a) In order to go into orbit, Phoebe must be going at least 17,500 mph. She took
enough fuel to last for 30 seconds. Will she orbit? Explain.
b) What is the minimum length of time that she could run and still get into orbit?
c) How long would the last stage have to run to get Phoebe going 25,000 mph so that
she could make it to the moon?
14. After you pour a cup of coffee, it cools off in such a way that the difference between
the coffee temperature and the room temperature decreases exponentially with time. This
model is called “Newton’s Law of Heating and Cooling.” Suppose that you pour a cup of
coffee. Three minutes later, you measure its temperature and find that it is 85C. Five
minutes after the first reading, you find that it has cooled to 72C. The room is 32C.
a) Write the equation the represents the rate at which the coffee is cooling.
b) What was the temperature of the coffee when it was first poured?
c) Assume that coffee is drinkable when its temperature is at least 55 C. When will
the coffee become undrinkable(colder)?
d) Plot the graph of the coffee’s temperature from the time the coffee was poured
until t=60. Where is the asymptote?
15. When rabbits were first brought to Australia in the 19th century, they had no natural
enemies, so their numbers increased rapidly. Assume that there were 65,000 rabbits in
1865, and that by 1867 the number had increased to 2,400,000. Assume that the number
of rabbits increased exponentially with the number of years that have elapsed since 1865.
a) How many rabbits would you predict were in Australia in 1870?
b) According to your model, when was the first pair of rabbits introduced into
Australia?