7-8 Using Exponential and Logarithmic Functions
1. PALEONTOLOGY The half-life of Potassium-40 is about 1.25 billion years.
a. Determine the value of k and the equation of decay for Potassium-40.
b. A specimen currently contains 36 milligrams of Potassium-40. How long will it take the specimen to decay to only
15 milligrams of Potassium-40?
c. How many milligrams of Potassium-40 will be left after 300 million years?
d. How long will it take Potassium-40 to decay to one eighth of its original amount?
SOLUTION: –kt
a. Exponential decay can be modeled by the function f (x) = ae , where a is the initial value, t is time in years, and k
is a constant representing the rate of continuous decay.
9
Substitute 1.25 × 10 , and 0.5a for t and f (x) respectively, then solve for k.
–10
b. Substitute 36, 15 and 5.545 × 10
for a, f (x) and k respectively then solve for t.
It will take 1,578,843,530 years.
c. Substitute 36, 300,000,000 and 5.545 × 10–10 for a, t and k respectively then evaluate.
After 300 million years, it will have about 30.48 mg of Potassium-40.
d. Substitute 36,
–10
and 5.545 × 10
for a, f (x) and k respectively then solve for t.
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After 300 million years, it will have about 30.48 mg of Potassium-40.
–10 Functions
7-8 d.
Using
Exponential
and Logarithmic
Substitute
36,
a, f (x) and k respectively then solve for t.
and 5.545
× 10 for
It will take 3,750,120,003 years.
2. SCIENCE A certain food is dropped on the floor and is growing bacteria exponentially according to the model y =
kt
2e , where t is the time in seconds.
a. If there are 2 cells initially and 8 cells after 20 seconds, find the value of k for the bacteria.
b. The “5-second rule” says that if a person who drops food on the floor eats it within 5 seconds, there will be no
harm. How much bacteria is on the food after 5 seconds?
c. Would you eat food that had been on the floor for 5 seconds? Why or why not? Do you think that the information
you obtained in this exercise is reasonable? Explain.
SOLUTION: a. Substitute 8 and 20 for y and t respectively, then solve for k.
b. Substitute 5 and 0.0693 for t and k respectively then solve for y.
It has 2.828 cells.
c. Sample answer: Yes; it has not even grown 1 cell in 5 seconds. There are many factors that affect this equation,
such as how clean the floor is and what type of food was dropped.
3. ZOOLOGY Suppose the red fox population in a restricted habitat follows the function
where t represents the time in years.
a. Graph the function for 0 ≤ t ≤ 200.
b. What is the horizontal asymptote?
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c. What is the maximum population?
d. When does the population reach 16,450?
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It has 2.828 cells.
Sample
answer: Yes;
has not even grown
1 cell in 5 seconds. There are many factors that affect this equation,
7-8 c.
Using
Exponential
andit Logarithmic
Functions
such as how clean the floor is and what type of food was dropped.
3. ZOOLOGY Suppose the red fox population in a restricted habitat follows the function
where t represents the time in years.
a. Graph the function for 0 ≤ t ≤ 200.
b. What is the horizontal asymptote?
c. What is the maximum population?
d. When does the population reach 16,450?
SOLUTION: a.
b. The horizontal asymptote is y = 16,500.
c. The maximum population is 16500.
d. Substitute 16450 for P(t) and solve fot t.
The population reaches 16450 in about 102 years.
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