Algebra 1
Exponential Decay Investigation
Name: _____________________________
When a large spill of a contaminant (like oil) is spilled into a large body of water,
authorities use boats called “skimmers” to filter the contaminant out of the water. A hose
is put into the water that sucks up many thousands of gallons of water every hour. The
water goes through an elaborate filter system and clean water is put back through another
hose. The oil that has been removed from the water is then dumped into another boat that
takes it away to be disposed of at a recycling center.
1. Imagine there was a large oil spill in Puget Sound. A skimmer was sent to the spill and
starts filtering the oil out of the water. Because of the size of the spill the boat will likely
spend over a week filtering the water.
a. Which day do you think the most oil would be filtered out?
b. Do you think the amount filtered out each day would increase or decrease as the week progressed?
c. In the end, would ALL the oil be filtered out, or would some still remain in the Sound?
d. Thinking about your answers to the above questions, which graph do you think would best
represent the amount of oil remaining in the sound each day? The x-axis is time and the yaxis is the amount of oil remaining in the water.
A
B
e. The data for the amount of oil remaining in the water each day is given below. Study the
table. What is the “decay factor”? In other words, what factor (number) is multiplied to get
from one day’s remaining gallons to the next day’s remaining gallons? (Note: Even though
there is a way to describe the change using division – we will not be using division during
this investigation)
Day Number
Gallons Remaining
0
2400
1
1200
2
600
3
300
4
150
5
75
6
37.5
7
f. Calculate and fill in the missing values in the table.
g. Write a Now/Next equation using multiplication for the above table. Include a start value.
h. Calculate how many gallons of oil are likely to be remaining on the 12th day.
i. Think back to your work with exponential growth. You wrote an equation y = a • b x , where
“a” is the start value and “b” is the growth factor. Exponential decay is the same except we
have a decay factor instead of a growth factor. Write an algebraic equation for this problem.
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2. The following table represents the height a ball bounces on consecutive bounces after
being dropped from 243 feet in the air.
Bounce Number
Height (in feet)
0
243
1
162
2
108
3
72
4
5
6
7
a. What is the “decay factor” in this problem? Again, in other words, what constant factor must
be continuously multiplied to get from one height value to the next height value?
b. Write the Now/Next equation and fill in the missing values on the table. Include a start value.
c. Write an algebraic equation for the bouncing ball. In this case our decay factor is a fraction,
so be sure that the exponent is on the outside of the parentheses.
d. Type your equation into the calculator and check your values on the table.
e. On what bounce does the ball first bounce less than 2 feet?
f. Produce a graph on your calculator for the
bouncing ball and sketch the graph to the right.
Remember to adjust your window settings to see
the graph.
g. When working with an exponential growth graph it was described as “increasing at an
increasing rate.” How should you describe an exponential decay graph?
h. For the following equations, 1) decide what situation the equation represents. Then 2) find
the answer. Make sure you use complete sentences.
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i)
" 2 %10
y = 243$ '
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ii)
" 2 %19
y = 243$ '
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iii)
" 2 %x
1 < 243$ '
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3. A ball is dropped from 25 feet and
4
ths of its height remains on each consecutive bounce.
5
a. Write a Now/Next equation for this situation. Include a start value.
€ this situation.
b. Write an algebra equation for
c. Fill in the given table of values for the first 5 bounces.
Bounce
Height (in ft)
0
1
2
3
4
5
d. On what bounce will the ball first bounce less than 1 ft?
4. Given the following table:
X
Y
0
1000
1
900
2
810
3
729
4
656.1
a. Write a Now/Next and an algebraic equation that models the data. Include a start value.
b. Type your equation into the y= part of the calculator and compare the values of the table. Are
they the same as the table above? If so, good for you! If they are not the same, try again.
c. What will the value of the table be at stage 30?
d. When will half of the original 1000 be left?
5. Given the following table:
X
Y
0
80
1
32
2
12.8
3
5.12
4
2.048
a. Write a Now/Next and an algebraic equation that models the data. Include a start value.
b. Type your equation into the y= part of the calculator and compare the values of the table. Are
they the same as the table above? If so, good for you! If they are not the same, try again.
c. At what stage will the value in the table be less than .1?
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6. Given the equation: y = 19683$ ' . Fill in the following table with the correct values.
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X
Y
0
1
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a. Write the Now/Next for this table.
b. What is the value at stage 10?
2
3
4
5
7. Given Next = Now x .8 (start at 50) fill in the following table.
X
Y
0
1
2
3
4
5
a. Write the algebra equation that matches this table.
b. At what stage will half of the original amount be left?
An important concept associated with exponential decay is the term half-life. Half-life is
defined as “how much time does it take for half of the original amount to remain?” This is
an important concept when discussing radioactive material and the absorption rate of
medication within the human body.
8. For instance, an important drug the body creates is insulin. Insulin is needed by the
body to maintain a person’s health. However, diabetics have trouble regulating the
amount of insulin in their body. Therefore, some diabetics must take insulin to assist
their body. Insulin breaks down rather quickly and the rate varies greatly depending
on the individual. The following table shows a typical pattern of the amount of insulin
in the body as time passes.
a. By looking at the graph, what
appears to be the half-life for insulin
in the case?
b. The pattern of decay shown on this
graph could be modeled using the
equation y = 10(0.95)x. What do the
values 10 and .95 mean in this
situation?
c. What would be the Now/Next
equation that would model this
situation?
d. To the nearest 10th of a minute, what is the half-life of insulin in this situation?
e. In the graph, the line is continuous since the breakdown of insulin is occurring as time
passes, not just at the end of each minute. Enter the algebraic equation into your calculator.
Then do the following: (use complete sentences.)
1) Tell what question the equation is asking AND
2) Then find the answer using your calculator.
i)
y = 10(0.95)1.5
ii)
y = 10(0.95)4.5
iii)
y = 10(0.95)18.75
iv)
y = 10(0.95)38
v)
6 < 10(0.95)x
vi)
2 > 10(0.95)x
f. Instead of having 10 units of insulin added to the body, a person had 50 units if insulin added
to the body. Adjust your algebraic equation and then find the half-life of this new situation.
9. One of the most famous drugs is penicillin. It was discovered in 1929, and it quickly
became known as the first “miracle drug,” because it was so effective in fighting
bacterial infections. While drugs react differently in each person, on average, a dose of
penicillin will be broken down by the body so that after each hour only 60% of the drug
remains that was present the previous hour. Suppose a person takes 300 milligrams of
penicillin at noon.
a. Write an algebraic equation that will model the amount of penicillin that will be in the
person’s bloodstream at the end of each hour.
Time
Amount of Penicillin (mg)
12
1pm
2pm
3pm
4pm
5pm
b. Fill in the following table.
c. To the nearest tenth of an hour, what is the half-life of penicillin in this specific case?
d. To the nearest tenth of an hour, when will there be less than 10mg of penicillin in the blood?
10. A different family of drugs that have been in the news for years are anabolic
steroids. These drugs have the ability to increase the weight and strength of the
individuals that take them. However, steroids can have very damaging effects on the
user. A certain steroid, Ciprionate, will retain 90% of itself in the user’s blood after
each 24-hour period. Suppose an individual takes 100 milligrams of Ciprionate.
a. Write an algebraic equation that will model the amount of Ciprionate that will be in the
person’s bloodstream at the end of each day.
b. To the nearest tenth of a day, what is the half-life of Ciprionate?
c. To the nearest tenth of a day, when will there be less than 5mg left in this person’s body?
11. From fuel for power plants to medical x-ray equipment to specific cancer treatments,
radioactive materials have become used in many important ways throughout the world.
However, radioactivity can also be very harmful, so these materials must be used
correctly and carefully. The radioactive chemical strontium-90 is produced during
many nuclear reactions. It decays very slowly, since about 98% of it will remain at the
end of each year.
a. If 100 grams of strontium-90 are present. How much of that radioactive material will be
around at the end of 1 year? After 2 years? After 3 years?
b. Write an algebraic equation that will model the amount of strontium-90 that will be present at
the end of each year?
c. Use your equation to find the amount of strontium-90 that will be present of the original
100grams, at the end of 20 years.
d. To the nearest tenth of year, find the half-life of strontium-90.