MATH 109

Exam 2 Review

JeopardyShow Me

The $$$$$Potent

PotablesFamous

Log CabinsCaptain’s

LogPotpourri

100 100 100 100 100

200 200 200 200 200

300 300 300 300 300

400 400 400 400 400

500 500 500 500 500

Potent Potables 100

• Suppose the amount of ozone in the atmosphere is decreasing exponentially at a continuous rate of 0.25% per year. How many years will it take for half the ozone to disappear?

• Answer: About 277 years

Potent Potables 200

• Plutonium-240 is a radioactive material which takes a long time to decay. If you have 100 grams initially, 1 year from now you’ll have 99.99 grams. Assuming that Plutonium-240 decays exponentially, what is its half-life?

• Answer: 6301 years

Potent Potables 300

• The graph of p(t), an exponential function, is shown:

• Find a formula for p(t)• Answer:

t

tp

5

4625

Potent Potables 400

• World population has been growing exponentially for the past 30 years. In 1987, the world population was 5 billion. In 1998, it was 6 billion. Determine the doubling time for the world’s population.

• Answer: About 42 years

Potent Potables 500

• Radioactive carbon-14 has a continuous decay rate of 0.0121% per year. A newly discovered skull contains 23% of its original carbon-14. Estimate the age of the skull. Round your answer to the nearest 1000 years.

• Answer: About 12000 years old.

Potpourri 100

• Let f(x) be given by the table below:

• Find k so that f(x) is (i) linear and (ii) exponential.• Answer: (i) k = 10 (ii) k = 6

x f(x)

0 2

1 K

2 18

Potpourri 200

• The cost of tuition at four-year public universities has been increasing roughly exponentially for the past several years. In 1997, average tuition was $3,111 while in 2004 it was $5,132, a growth of about 65% over a 7 year period. Assuming this trend continues every 7 years, at what rate is tuition increasing each year.

• Answer: 7.41%

Potpourri 300

• Consider the graphs and equations for the exponential functions shown below:

• Rank the decay rates from smallest to largest in magnitude.• Answer: r, b, k

Potpourri 400

• Suppose a dead body cools according to the equation below, where t is measured in hours after death and H is the temperature of the body at that time. What is the temperature of the body 5 hours after death?

• Answer: 82 degrees

23

75log10

Ht

Potpourri 500

• The population of the small Asian state of Tuva was 200000 in the year 1992 and 300,000 in the year 2002. Assuming the population is growing exponentially, in what year will it pass 350,000?

• Answer: Around 2006

Captain’s Log 100

• Evaluate the following WITHOUT a calculator:

100log10

10loglog

100log100log

53 101010log

100

0

12

30

31

Captain’s Log 200

• Simplify WITHOUT a calculator:

3log1000

2log01.0

1010log/1log

1

27

4

1

1

Captain’s Log 300

• Simplify the following completely:

xex ln2ln

xln2

3

Captain’s Log 400

• Simplify the following completely:

eeAA

ln

3

log

loglog

Aln2

5

Captain’s Log 500

• Given the following:

• Find:

• Answer: 7.5

5log Ab 11log Db 3log Qb

2

logQ

ADb

Famous Log Cabins 100

• Simplify each expression:

x

BA

A

ee

e

e

4log

ln

2

1ln

21

2

2

ln3ln2

5

x

BAee

Ae

e

x

BA

A

4

12

32ln3ln2

55

4log

2ln

21ln

21

2

21

Famous Log Cabins 200

• Expand the following logarithm completely:

25

10log

ab

baab 55525 log2log12log10

log

Famous Log Cabins 300

• Write the following as a single logarithm:

8lnlnln23

1 yx

2ln8lnlnln2

3

1 31

32

yxyx

Famous Log Cabins 400

• Which of the following is/are true?

cMc

M

ee

eeeee

e

1

lnln

12lnln2ln2

ln

2lnlnln2lnln2lnln

2ln

Famous Log Cabins 500

• Solve for x using logs:

• Answer: about 2.5

235002000x

Show Me The Money 100

• How much money should you invest at 6% compounded quarterly so that you have $20,000 after 9 years?

• Answer:$11,701.79

Show Me The Money 200

• Suppose $10,000 was invested in a trust fund for a child. After 20 years, the fund has matured to a value of $27,126.40. What is the interest rate of the fund if interest has been compounded continuously?

• Answer: 4.99%

Show Me The Money 300

• You decide to invest $10000 in a college trust fund for your newborn child at 9% interest compounded continuously for 18 years. You learn of another trust fund which compounds quarterly. What nominal rate does the second trust fund need to be in order for both investments to be equally good for you?

• Answer: 9.1%

Show Me The Money 400

• How long does it take for a $10,000 US Treasury Bond earning interest at 2.4% compounded monthly to mature to $15,000?

• Answer: Around 17 years

Show Me The Money 500

• How long does it take for an investment of $5000 to double if it is invested at a rate of 7.5% compounded quarterly?

• Answer: Around 9 years