Checkup: Applications of Logarithms
Answer the following questions using what you've learned from this lesson. Write your responses in the space provided, and turn the assignment in to your instructor.
For 1 – 2, use the following situation:
In 1991, the cost of mailing a 1 oz. first-class letter was 29 cents, and the inflation rate was 4.6%. If the inflation rate stayed constant, the function C(t) = .29(1.046)t would represent the cost of mailing a first-class letter as a function of years since 1991.
1. If the function given holds true, in what year would the cost of mailing a first-class letter reach 60 cents?
The cost of mailing a first-class letter would reach 60 cents in 2007.
2. In 2007, the cost of mailing a first-class letter was 41 cents. Has the inflation rate stayed constant since 1991? Explain.
No the inflation rate has not remained constant because in 1991 the cost of mailing a first-class letter was 29 cents, causing the inflation rate to be 4.6%. If this rate were to be constant until 2007, the cost would be 60 cents instead of 41 cents.
For 3 – 5, use the following situation:
A hot bowl of soup cools according to Newton’s law of cooling. Its temperature (in degrees Fahrenheit) at time t is given by T(t) = 68 + 144e-.04t, where t is given in minutes.
3. What was the initial temperature of the soup?
212 degrees Fahrenheit
4. What is the temperature of the soup after 15 minutes?
147.03 degrees Fahrenheit
5. How long after serving is the soup 125F?
6. A forensic detective is called to the scene of a murder at 2 a.m. When she checks the temperature of the body, she finds that it is 80F. The temperature of the room in which the body is found is 65F.
If the detective uses Newton’s law of cooling, T(t) = TA + (T0 - TA))e-0.1947t, what will she say was the time of death?
7. An archeologist finds an artifact that contains 44% of the carbon-14 that it would have had originally. What is the estimated age of the artifact? (The half-life of carbon-14 is 5,730 years.)
Approximately 6, 787 years
For 8 – 9, use the following situation:
A population of bacteria begins with 1500 bacteria and grows to 4500 in one hour.
8. Find a function that represents the growth of this culture of bacteria as a function of time.
The bacteria becomes 3 fold in 1 hour.
N = 1500(3t)
9. How long does it take this culture of bacteria to double?
3000 = 1500(3t)
2 = 3t
T(log 3) = log 2
t = log 2/log 3 hours = 60(log 2)/log 3
≈ 37.86 min
For 10 – 13, use the following situation:
A potato is put into an oven that has been heated to 350F. Its temperature as a function of time is given by T(t) = a(1 - e-kt) + b. The potato was 50F when it was first put into the oven.
10. What is the value of b in this context? Explain.
B is the constant and 50F is the constant, so:
b = 50(e-kt)
11. What is the value of a in this context? Explain.
You are multiplying the whole function by 350, which is also “a” purpose, therefore: a = 350
12. If the potato is 60F after 2 minutes, what is the value of k? Explain.
T (0) = 50ºF = a (1-e-k (0)) +b
T (2) = 60ºF = a (1-e-k (2)) +b
50−b/a= 1 − e0
60−b/a= 1 – e-2k
13. When will the potato reach 150F? Explain.
2 minutes – 60F
4 minutes – 120F
4.5 minutes – 150F (halt of 60 is thirty, so in 4.5 minutes is should reach 150F)
In 4.5 minutes.
For 14 – 17, use the following situation:
The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function:
N(t) = 10,000/(5 + 1245e-.97t)
14. How many people were in the group that started the rumor?
15. How many people have heard the rumor after 1 day? After 5 days?
Day 1: approx. 21
Day 5: approx. 678
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