7.6c Exponential Decay Investigation
Algebra
name _______________________________ per __
Activity 1: Oil Spill
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 y-axis is the amount of oil
remaining in the water.
B
A
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
f.
Calculate and fill in the missing values in the table.
g.
Write a recursive equation using multiplication for the above table. Include a start value.
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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 · bx , 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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Activity 2: Bouncing Ball
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)
a.
0
243
1
162
2
108
3
72
4
5
6
7
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 recursive 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. Write your scale on 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.
i)
æ 2 ö10
y = 243ç ÷
è 3ø
ii)
æ 2 ö19
y = 243ç ÷
è 3ø
iii)
æ 2 öx
1 < 243ç ÷
è 3ø
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3. A ball is dropped from 25 feet and
a.
4
ths of its height remains on each consecutive bounce.
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Write a recursive equation for this situation. Include a start value.
b. Write an algebra equation for this situation.
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.
A ball is dropped from 40 feet and each time it bounces, its rebound is 25% lower than it was the
bounce before.
a.
Write a recursive equation for this situation. Include a start value.
b. Write an algebra equation for this situation.
c.
How is this situation different from the previous ones?
Activity 4: Generalizing
4. Given the following table:
a.
X
Y
0
1000
1
900
2
810
3
729
4
656.1
Write a recursive 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:
a.
X
Y
0
80
1
32
2
12.8
3
5.12
4
2.048
Write a recursive 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:
X
Y
a.
æ 1 öx
y = 19683ç ÷ .
è 3ø
0
Fill in the following table with the correct values.
1
2
3
4
5
3
4
5
Write the recursive equation for this table.
b. What is the value at stage 10?
7. Given ANS = LAST x .8 (start at 50) fill in the following table.
X
Y
a.
0
1
2
Write the algebra equation that matches this table.
b. At what stage will half of the original amount be left?
Activity 5: Half-Life
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 recursive equation that
would model this situation?
d. To the nearest 10th of a minute, what is the half-life of insulin in this situation?
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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. You should have 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.
f.
i)
y = 10(0.95)1.5
ii)
y = 10(0.95)38
iii)
6 < 10(0.95)x
iv)
2 > 10(0.95)x
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.
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?
Time
Amount of Penicillin (mg)
12
1pm
2pm
3pm
4pm
5pm
d. To the nearest tenth of an hour, when will there be less than 10mg of penicillin in the blood?
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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.
Write an algebraic equation that will model the amount of strontium-90 that will be present at the end of
each year?
b. 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?
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.
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