Math 8
Mathematics homework help
Lesson 4.4
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).
· 24. Translate exponential statements to equivalent logarithmic statements, interpret logarithmic scales, and use logarithms to solve basic exponential equations.
· 25. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand
· That log scales reveal magnitudes and relative change
Students will be able to
· Solve applied problems using logarithms
· Read and place values on a logarithmic scale
In the last lesson, you learned how to solve exponential equations. In this lesson we will explore some applications of that skill. We will also look at how logarithms are useful for representing numbers that vary greatly in size.
Carbon Dating
Problem Situation 1: Carbon Dating
When ancient artifacts are found, scientists determine their age using a technique called radioactive dating. Carbon-14 is a radioactive isotope that is in living organisms. All living organisms have approximately the same ratio of carbon-14 to carbon-12. After the organism dies, the carbon-14 decays to nitrogen-14, changing the ratio of carbon-14 to carbon-12. The fraction of the original carbon-14 remaining in an organism after t thousand years is modeled by A=0.8861tA=0.8861t .
#1 Points possible: 12. Total attempts: 5
Suppose an object, such as a tree, has 1 unit of carbon-14 in it at the time it dies. Complete the table for the fraction of carbon-14 remaining after the given numbers of years since death. Give your values to 3 decimal places.
| Years since death | Carbon-14 Remaining | |
| 0 | 1.000 | |
| 2000 | ||
| 4000 | ||
| 6000 | ||
| 8000 |
Hint: Be careful with how t is defined in the equation. t represents thousands of years, so t = 1 is one thousand years.
#2 Points possible: 5. Total attempts: 5
Textbooks will commonly report that the “half-life” of carbon-14 is 5730 years. Based on your calculations, what do you think “half-life” means?
· After 5730 years, half the carbon-14’s life will have passed, after which it will grow back to its original amount
· Carbon-14 has a total life of 11460 years, after which it will be gone
· The amount of carbon-14 remaining decreases by half in 5730 years, then after another 5730 years there will be half of that remaining, or 25% of the original
· The amount of carbon-14 remaining decreases by half in 5730 years, then after another 5730 years there will be none remaining
#3 Points possible: 12. Total attempts: 5
Just using the idea of half-life, complete the table.
| Years since death | Carbon-14 Remaining |
| 0 | 1.000 |
| 5730 | |
| 11460 | |
| 17190 | |
| 22920 |
#4 Points possible: 5. Total attempts: 5
An artifact is found, and scientists determine that it contains 20% of its original carbon-14.
Estimate the age of the artifact, using the results from the previous problem.
years old
#5 Points possible: 5. Total attempts: 5
An artifact is found, and scientists determine that it contains 20% of its original carbon 14.
Determine the approximate age of the artifact using algebra and the equation A=0.8861tA=0.8861t given at the start of the problem situation. Give your answer as a number of years, to the nearest year.
years old
Log Scales
Problem Situation 2: Log Scales
When numbers are very small or very large, or a set of values varies greatly in size, it can be hard to represent those values. Consider the distance of the planets of our solar system from the sun:
| Planet | Distance (millions of km) |
| Mercury | 58 |
| Venus | 108 |
| Earth | 150 |
| Mars | 228 |
| Jupiter | 779 |
| Saturn | 1430 |
| Uranus | 2880 |
| Neptune | 4500 |
Placed on a linear scale – one with equally spaced values – these values get bunched up.
However, representing each value as a power of 10, and using a scale spaced by powers of 10, the values are more reasonably spaced.
#6 Points possible: 5. Total attempts: 5
The scale above shows the dwarf planet Pluto. Use the scale to estimate the distance in kilometers from the sun to Pluto. Round to the nearest million km.
million km
#7 Points possible: 5. Total attempts: 5
Ida is a large asteroid in asteroid belt, approximately 429 million kilometers from the sun. Plot on the logarithmic scale below where Ida belongs.
101.5
102
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104
Clear All Draw: Dot
Vertical Scale
Sometimes you will see a logarithmic scale on the vertical axis of a graph. A common example is in stock charts. Both stock charts below show the Dow Jones Industrial Average, from 1928 to 2010.
Both charts have a linear horizontal scale, but the first graph has a linear vertical scale, while the second has a logarithmic vertical scale.
In 1929, the stock market value dropped from 380 to 42. In 2008, the stock market value dropped from 14,100 to 7,500.
#8 Points possible: 16. Total attempts: 5
Compute the absolute and relative change in each market drop. Give your answers to 1 decimal place. Since we’re looking for the size of the drop, give your answers as positive values.
| 1929: | Absolute change: | Relative Change: % |
| 2008: | Absolute change: | Relative Change: % |
#9 Points possible: 10. Total attempts: 5
Which graph better reveals the drop with the largest absolute change?
· The first graph, with linear scale
· The second graph, with log scale
Which graph better reveals the drop with the largest relative change?
· The first graph, with linear scale
· The second graph, with log scale
#1 Points possible: 5. Total attempts: 5
Polluted water is passed through a series of filters. Each filter removes 90% of the remaining impurities from the water. If you have 10 million particles of pollutant per gallon originally, how many filters would the water need to be passed through to reduce the pollutant to below 500 particles per gallon? You can only use a whole number of filters.
filters
#2 Points possible: 5. Total attempts: 5
India is the second most populous country in the world, with a population in 2008 of about 1.14 billion people. The population is growing by about 1.34% each year. If the population continues following this trend, during what year will the population reach 2 billion?
#3 Points possible: 5. Total attempts: 5
If $1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to $1500? Round to the nearest month.
years, months
#4 Points possible: 5. Total attempts: 5
In Lesson 4.2, you found a model for a spread of a disease.
Suppose that another outbreak has been detected. On Sept 1, there were 60 reported cases. A week later, there were 75 reported cases.
If the disease continues spreading exponentially, how many weeks after Sept 1 will 850 cases be expected? Round to the nearest week.
weeks
#5 Points possible: 5. Total attempts: 5
In chemistry, the acidity of a liquid is commonly measured in pH. pH is important outside of chemistry to many things, including keeping swimming pools maintained, and measuring the acidity of rain. The pH scale is related to the concentration of hydrogen ions in the liquid, a number that tends to be very small, so the pH scale was introduced to make the numbers easier to work with. For a given concentration of hydrogen ions, H, measured in Molars, the pH is determined by pH = -log(H)
Purified tap water has a hydrogen ion concentration of 10-7 M. Find the pH of water.
#6 Points possible: 5. Total attempts: 5
The number line below shows a value on a log scale.
10-5
10-4
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10-1
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What is, approximately, the value of the point shown on the number line? x =
#7 Points possible: 5. Total attempts: 5
Plot the number 0.0001 on the log scale below.
10-5
10-4
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10-1
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Clear All Draw: Dot
#8 Points possible: 5. Total attempts: 5
Plot the number 0.0032 on the log scale below.
10-5
10-4
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10-1
100
101
102
103
104
105
Clear All Draw: Dot