Exponential and Logarithmic Modeling
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. An initial population of 505 quail increases at an annual rate of 23%. Write an exponential function to model
the quail population.
a.
b.
c.
d.
____
2. Write an exponential function
a.
b.
c.
____
3. Find the annual percent increase or decrease that
models.
a. 230% increase b. 130% increase c. 30% decrease d. 65% decrease
____
4. The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801
kg. Write an exponential function that models the decay of this material. Find how much radioactive material
remains after 10 days. Round your answer to the nearest thousandth.
a.
d.
; 0.228 kg b.
for a graph that includes (1, 15) and (0, 6).
d.
; 0 kg c.
; 738.273 kg
; 0.911 kg
____
5. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will you
have in the account after 4 years?
a. $800.26 b. $6,701.28 c. $10,138.07 d. $1,923.23
____
6. How much money invested at 5% compounded continuously for 3 years will yield $820?
a. $952.70 b. $818.84 c. $780.01 d. $705.78
____
7. The table shows some notable earthquakes that occurred in recent years. How many times more energy was
released by the earthquake in Peru than by the earthquake in Mexico?
Earthquake Location
Date
Richter Scale Measure
Italy
October 31, 2002
5.9
El Salvador
February 13, 2001
6.6
Afghanistan
May 30, 1998
6.9
Mexico
January 22, 2003
7.6
Arequipa, Peru
June 23, 2001
8.1
[Source: World Almanac 2004, p. 190]
a. about 0.50 times as much energy b. about 15 times as much energy c. about 37.52 times as much energy
d. about 5.48 times as much energy
The pH of a liquid is a measure of how acidic or basic it is. The concentration of hydrogen ions in a liquid is labeled
. Use the formula
to answer questions about pH.
____
8. Find the pH level, to the nearest tenth, of a liquid with [H+] about
a. –3.8 b. 3.8 c. 2.2 d. 3.0
____
9. A construction explosion has an intensity I of
.
W/m2. Find the loudness of the sound in decibels if
and
W/m2. Round to the nearest tenth.
a. 146.9 decibels b. 115.8 decibels c. 106.9 decibels d. 48.5 decibels
____ 10. A company with loud machinery needs to cut its sound intensity to 37% of its original level. By how many
decibels would the loudness be reduced? Use the formula
. Round to the nearest hundredth.
a. 2.01 decibels b. 2.12 decibels c. 1.37 decibels d. 4.32 decibels
____ 11. Solve
.
a. 0.0090 b. 0.3103 c. 3.2222 d. 111
____ 12. Solve
.
a. 12.3308 b. 43.3013 c. 86.6025 d. 1875
Write the expression as a single natural logarithm.
____ 13.
a.
b.
c.
d.
____ 14.
a.
b.
c.
d.
____ 15. The sales of lawn mowers t years after a particular model is introduced is given by the function y =
, where y is the number of mowers sold. How many mowers will be sold 2 years after a model
is introduced? Round the answer to the nearest whole number.
a. 37,897 mowers b. 7,383 mowers c. 15,901 mowers d. 17,000 mowers
____ 16. The generation time G for a particular bacteria is the time it takes for the population to double. The bacteria
increase in population is shown by the formula
, where t is the time period of the population
increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the
end of the time period. If the generation time for the bacteria is 6 hours, how long will it take 8 of these bacteria to multiply into a colony of 7681 bacteria? Round to the nearest hour.
a. 177 hours b. 76 hours c. 4 hours d. 85 hours
____ 17. The amount of money in an account with continuously compounded interest is given by the formula
, where P is the principal, r is the annual interest rate, and t is the time in years. Calculate to the
nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded
continuously at 6.2%. Round to the nearest tenth.
a. 1.1 yr b. 6.9 yr c. 11.2 yr d. 0.6 yr
Short Answer
18. Without graphing, determine whether the function
decay.
represents exponential growth or exponential
19. Without graphing, determine whether the function
decay.
represents exponential growth or exponential
20. The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10
years.
Value($)
9000
8000
7000
6000
5000
4000
3000
2000
1000
1
2
3
4
5
6
7
8
9 10 11 years
a. Write an exponential function for the graph.
b. Use the function in part a to find the value of the boat after 9.5 years.
21. One method of dating artifacts is radiocarbon dating. An artifact’s age y in years can be found by using the
function
, where x is the number of beta radiation emissions per minute per
gram of carbon in the artifact.
a. Use a graphing calculator to graph the function. Sketch the graph on the grid.
y
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
1
2
3
4
5
6
7
8
9
10
x
b. Use your calculator to find the approximate value of y when x is 6.
Essay
22. The table shows the number of squirrels in a particular forest t years after a forest fire.
Number of Squirrels
Years
Squirrels
0
30
1
60
2
120
3
240
4
480
5
960
a. Explain how the population of squirrels is changing each year.
b. Write a function to model the situation. Explain what each number represents.
23. Suppose you invest $580 at 10% compounded continuously.
a. Write an exponential function to model the amount in your investment account.
b. Explain what each value in the function model represents.
c. In how many years will the total reach $3600? Show your work.
24. The formula
gives the average atmospheric pressure P in pounds per square inch, at an altitude x in miles above sea level.
a. Find the elevation at which the average atmospheric pressure is 8.4 lb/in. 2. Show the steps you used to
solve this problem.
b. What is the average atmospheric pressure at sea level? Explain.
Other
25. In a particular region of a national park, there are currently 330 deer, and the population is increasing at an annual rate of 11%.
a. Write an exponential function to model the deer population.
b. Explain what each value in the model represents.
c. Predict the number of deer that will be in the region after five years. Show your work.
Exponential and Logarithmic Modeling
Answer Section
MULTIPLE CHOICE
1. ANS: D
PTS: 1
DIF: L1
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.1 Exponential Growth
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 2
KEY: exponential function | growth factor
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
2. ANS: A
PTS: 1
DIF: L1
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.1 Exponential Growth
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 3
KEY: exponential function | growth factor
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
3. ANS: B
PTS: 1
DIF: L3
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.2 Exponential Decay
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 6
KEY: exponential decay | exponential function | exponential growth | percent
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
4. ANS: C
PTS: 1
DIF: L1
REF: 8-2 Properties of Exponential Functions
OBJ: 8-2.1 Comparing Graphs
NAT: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
STA:
NC 2.03
TOP: 8-2 Example 3
KEY: exponential decay | exponential function
MSC: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
5. ANS: D
PTS: 1
DIF: L1
REF: 8-2 Properties of Exponential Functions
OBJ: 8-2.2 The Number e
NAT: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
6.
7.
8.
9.
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
STA:
NC 2.03
TOP: 8-2 Example 5
KEY: exponential function | exponential growth | interest rates | problem solving | the number e | compounding continuously | percent
MSC: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
ANS: D
PTS: 1
DIF: L2
REF: 8-2 Properties of Exponential Functions
OBJ: 8-2.2 The Number e
NAT: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
STA:
NC 2.03
KEY: exponential function | exponential growth | interest rates | percent | problem solving | the number e |
compounding continuously
MSC: NAEP A1e | NAEP A1h | CAT5.LV21/22.45 | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM
| IT.LV17/18.DI | IT.LV17/18.DP | IT.LV17/18.PS | S9.TSK3.DSP | S9.TSK3.NS | S9.TSK3.PRA |
S10.TSK3.DSP | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 | TV.LV21/22.47 |
TV.LV21/22.52 | TV.LVALG.56
ANS: D
PTS: 1
DIF: L1
REF: 8-3 Logarithmic Functions as Inverses
OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions
NAT: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-3 Example 1
KEY: problem solving | evaluating logarithms | logarithm
MSC: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
ANS: C
PTS: 1
DIF: L2
REF: 8-3 Logarithmic Functions as Inverses
OBJ: 8-3.1 Writing and Evaluating Logarithmic Expressions
NAT: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-3 Example 4
KEY: logarithm | problem solving
MSC: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
ANS: C
PTS: 1
DIF: L1
REF: 8-4 Properties of Logarithms
OBJ: 8-4.1 Using the Properties of Logarithms
NAT: NAEP A2e | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.GM
| S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 | TV.LV21/22.52 | TV.LVALG.53
STA:
NC 1.01
TOP: 8-4 Example 4
KEY: properties of logarithms | problem solving
MSC: NAEP A2e | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.GM
| S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 | TV.LV21/22.52 | TV.LVALG.53
10. ANS: D
PTS: 1
DIF: L1
REF: 8-4 Properties of Logarithms
OBJ: 8-4.1 Using the Properties of Logarithms
NAT: NAEP A2e | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.GM
| S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 | TV.LV21/22.52 | TV.LVALG.53
STA:
NC 1.01
TOP: 8-4 Example 4
KEY: properties of logarithms | problem solving
MSC: NAEP A2e | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.GM
| S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 | TV.LV21/22.52 | TV.LVALG.53
11. ANS: A
PTS: 1
DIF: L2
REF: 8-5 Exponential and Logarithmic Equations
OBJ: 8-5.2 Solving Logarithmic Equations
NAT: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
STA:
NC 1.01 | NC 2.01
TOP:
8-5 Example 7
KEY: logarithmic equation | properties of logarithms
MSC: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
12. ANS: B
PTS: 1
DIF: L2
REF: 8-5 Exponential and Logarithmic Equations
OBJ: 8-5.2 Solving Logarithmic Equations
NAT: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
STA:
NC 1.01 | NC 2.01
TOP:
8-5 Example 7
KEY: logarithmic equation | properties of logarithms
MSC: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
13. ANS: D
PTS: 1
DIF: L1
REF: 8-6 Natural Logarithms
OBJ: 8-6.1 Natural Logarithms
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-6 Example 1
KEY: simplifying a natural logarithm | properties of logarithms
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
14. ANS: D
PTS: 1
DIF: L2
REF: 8-6 Natural Logarithms
OBJ: 8-6.1 Natural Logarithms
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-6 Example 1
KEY: simplifying a natural logarithm | properties of logarithms
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
15. ANS: D
PTS: 1
DIF: L1
REF: 8-6 Natural Logarithms
OBJ: 8-6.1 Natural Logarithms
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-6 Example 2
KEY: simplifying a natural logarithm | logarithmic function | problem solving
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
16. ANS: D
PTS: 1
DIF: L2
REF: 8-5 Exponential and Logarithmic Equations
OBJ: 8-5.1 Solving Logarithmic Equations
NAT: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
STA:
NC 1.01 | NC 2.01
TOP:
8-6 Example 2
KEY: evaluating logarithms | logarithmic equation | properties of logarithms | problem solving
MSC: NAEP A2b | CAT5.LV21/22.50 | CAT5.LV21/22.51 | IT.LV17/18.AM | IT.LV17/18.CP | S9.TSK3.NS |
S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.56
17. ANS: C
PTS: 1
DIF: L2
REF: 8-6 Natural Logarithms
OBJ: 8-6.2 Natural Logarithmic and Exponential Equations
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-6 Example 5
KEY: exponential equation | properties of logarithms | problem solving
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
SHORT ANSWER
18. ANS:
exponential growth
PTS: 1
DIF: L1
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.2 Exponential Decay
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 4
KEY: exponential function | exponential growth | reasoning
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
19. ANS:
exponential decay
PTS: 1
DIF: L1
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.2 Exponential Decay
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 4
KEY: exponential decay | exponential function | reasoning
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
20. ANS:
a. Estimates for a and b may vary. Sample:
b. about $250
PTS: 1
DIF: L1
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.2 Exponential Decay
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 6
KEY: exponential decay | exponential function | graphing | problem solving | multi-part question
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
21. ANS:
y
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
1
2
3
4
5
6
7
8
9
10
x
a.
b. about 6,827 years
PTS: 1
DIF: L3
REF: 8-3 Logarithmic Functions as Inverses
OBJ: 8-3.2 Graphing Logarithmic Functions
NAT: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-3 Example 6
KEY: graphing | logarithmic function | multi-part question | problem solving
MSC: NAEP A2a | NAEP A3a | NAEP A3b | CAT5.LV21/22.50 | CAT5.LV21/22.55 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.GM | S9.TSK3.PRA | S10.TSK3.GM | S10.TSK3.PRA | TV.LV21/22.13 |
TV.LV21/22.52 | TV.LVALG.56
ESSAY
22. ANS:
[4] a. Answers may vary. Sample: The population doubles each year. It appears that the population is increasing exponentially.
b. A function for the situation is
. The number 30 represents the initial population at 0 years.
The number 2 represents the growth factor of 2.
[3] one minor mathematical error or a reasoning error
[2] poor explanation for part a. or error in the function model
[1] only one part of the question answered
PTS: 1
DIF: L2
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.1 Exponential Growth
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 2
KEY: exponential function | growth factor | problem solving | writing in math | extended response | rubric-based question
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
23. ANS:
[4] a.
b. In the model, the coefficient of e is 580, the original investment. The formula for continuously compounded interest uses the number e raised to the power rt, where r is the rate as a decimal, in this case
0.1, and t is the time in years.
c. To find the number of years to reach $3600, substitute 3600 into the model.
Dividing and rounding to the nearest year, t  18. The investment will reach $3600 in about 18 years.
[3] one error in computation or incomplete explanation
[2] two errors in computation or no explanation
[1] one correct answer with no explanation
PTS: 1
DIF: L3
REF: 8-6 Natural Logarithms
OBJ: 8-6.2 Natural Logarithmic and Exponential Equations
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
KEY: compounding continuously | exponential function | exponential growth | extended response | graphing |
interest rates | percent | problem solving | the number e | writing in math | rubric-based question
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
24. ANS:
[4] a.
Substitute 8.4 for P.
Divide each side by 14.7.
Take the natural logarithm of each side.
Simplify.
Divide each side by –0.21.
Use a calculator.
The elevation is about 2.7 miles above sea level.
b. The average atmospheric pressure at sea level is 14.7 lb/in. 2 because x is 0 and
[3] one mathematical error or one incorrect answer
[2] two mathematical errors or one error and an incomplete explanation
[1] one correct answer with no explanation
.
PTS: 1
DIF: L3
REF: 8-6 Natural Logarithms
OBJ: 8-6.2 Natural Logarithmic and Exponential Equations
NAT: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 1.01 | NC 2.01
TOP: 8-6 Example 5
KEY: exponential equation | properties of logarithms | problem solving | extended response | rubric-based
question | writing in math
MSC: NAEP A3a | CAT5.LV21/22.50 | CAT5.LV21/22.51 | CAT5.LV21/22.52 | IT.LV17/18.AM |
IT.LV17/18.CP | S9.TSK3.NS | S9.TSK3.PRA | S10.TSK3.NS | S10.TSK3.PRA | TV.LV21/22.11 |
TV.LV21/22.12 | TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
OTHER
25. ANS:
a.
b. In the model, 330 represents the initial population of deer. The growth factor is represented by 1 + 0.11 or
1.11.
c. To predict the number of deer present after 5 years, substitute 5 into the function and evaluate.
function model
Substitute 5 for x.
Use a calculator.
There will be about 556 deer in the region.
PTS: 1
DIF: L2
REF: 8-1 Exploring Exponential Models
OBJ: 8-1.2 Exponential Decay
NAT: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56
STA: NC 2.03 | NC 2.03b | NC 2.04 | NC 2.04b
TOP: 8-1 Example 2
KEY: exponential decay | exponential function | multi-part question | percent | problem solving | writing in
math
MSC: NAEP A2h | CAT5.LV21/22.50 | CAT5.LV21/22.53 | IT.LV17/18.AM | IT.LV17/18.DI | IT.LV17/18.PS
| S9.TSK3.DSP | S9.TSK3.PRA | S10.TSK3.DSP | S10.TSK3.PRA | TV.LV21/22.15 | TV.LV21/22.17 |
TV.LV21/22.52 | TV.LVALG.53 | TV.LVALG.56