Pre-AP Algebra 2 Unit 9 - Lesson 9 Using a logarithmic scale to model the distance between planets and the Sun.

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1 Pre-AP Algebra 2 Unit 9 - Lesson 9 Using a logarithmic scale to model the distance between planets and the Sun. Objectives: Students will be able to read a graph with a logarithmic scale. Students will be able to use formulas for decibels and magnitudes to determine and compare loudness and intensity of sound waves and earthquakes. Materials: Hw #9-8 answers packet; Do Now, pair work, homework #9-9 Time Activity 5 min Homework Check Hand out answers packets to hw #9-8. Pass around tally sheets. 10 min Homework Review Present the solutions to the top 2-3 problems from the homework, based on the tally sheets. 10 min Do Now Hand out the Do Now sheet with the planets data. Ask students to graph the distances of the planets on the pair of axes. Rule: all data points must be included. After students work on this for a few minutes, discuss some of the problems that they had completing the task. The fact that the numbers are so far apart makes it difficult. If the scale is too small, then you can t get all the data points on the graph. If the scale is too big, many of the data points are squished so far together that you can t tell them apart. We are going to use logarithms today to resolve these problems. 25 min Direct Instruction Ask students to turn to the back of the Do Now sheet. Label the second column log d and then calculate the log of each distance. Once this is done, notice that the logs are much closer together. Label the y-axis log d, create a good scale, and then make a bar graph. Discuss the benefits of graphing with a log scale. Highlight the fact that each square on the y-axis now represents a different amount, even thought the scale is constant. Moving up by a square is the same as multiplying the underlying value by a factor of 10. Explain that these scales are used in several key realworld applications: distance, magnitude of earthquakes, and loudness of sound. 30 min Pair Work Hand out the Earthquakes handout to students to practice on. Review answers on the overhead. Homework #9-9:

2 Pre-AP Algebra 2 Do Now Name: Logs what are they good for? The planets in the Milky Way galaxy orbit the sun at different distances 1. Using the axes given, scale the y-axis so that all the points given in the table can be plotted on the graph. Remember every square on the y-axis must be worth the same amount. Planet Distance from Sun Mercury 36 Venus 68 Earth 92 Mars 141 Jupiter 484 Saturn 888 Uranus 1,783 Neptune 2,799 Pluto 3,67 1 Distances are in millions of miles so 36 means 36 million miles.

3 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Follow-up questions. 1.) Milo is in orbit around the Sun at a distance of 630 million miles from the Sun. What would the scaled value of this distance be?

4 2.) Otis is in orbit around the Sun at a distance that has a scaled value of 4.9. How many miles from the Sun is Otis? For this question you need to divide Milo s distance (in million miles) by Otis s distance (in million miles). For example, since Saturn is 888 million miles away from the Sun and Earth is 92 million miles away from the sun, Saturn is 9.65 times further ) How many times is Milo further than Otis from the Sun?

5 San Francisco 1906 Tokyo 1923 Chile 1960 China 1976 Indonesia 2004 Loma Prieta 1989 El Salvador 2001 India 2001 Los Angeles 1994 Pre-AP Algebra Pair Work Name: EARTHQUAKES! Earthquakes can be measured using seismographs that can detect vibrations as little as 0.001mm in the ground. These readings can be logarithmically converted into magnitudes and the Richter scale compares these magnitudes. This is the function: M( R) log( R) 3 where M is the magnitude and R is the seismic reading in mm. 1. First finish the table 2. Make a bar graph on the axes below Location M(R) R San 8 Francisco Tokyo 8.25 Chile 9 China 31,623 Indonesia 1,000,000 Loma 6.9 Prieta El Salvador India 9.1 Los Angeles 3,981 5,981

6 1. In 1984 there was an earthquake in Mexico City that measured 8.1 on the Richter scale. Ten years later there was an earthquake in Los Angeles that measured 6.6 on the Richter scale. In order to accurately compare these two earthquakes do the following: a. Find the seismographic reading for the earthquake in Mexico. b. Find the seismographic reading for the earthquake in LA. c. Using the seismographic readings, find how many times more intense the Mexico City earthquake was than the Los Angeles. 2. Last weekend, a meteorite crashed to the Earth and registered a seismic reading of 398 mm. Compare the intensity of the meteorite crash to that of the 8.2 magnitude earthquake that happened in Chile in 1960 by finding the ratio of their seismographic readings.

7 Softest noise that can be heard Whisper across room Light rainfall Normal conversation Car driving, heard from 50 ft away Noisy factory Subway train Rock concert Shotgun blast Pre-AP Algebra 2 Homework #9-9 Name: HW #9-9: Logarithmic Scales LOUDNESS! In class, you learned about how logarithms can be used to more easily plot out and work with large distances and the intensity of earthquakes. Another application of logarithms is to deal with how LOUD a sound is. To do this, we have developed a unit called decibels, which is a measure of the loudness of a sound. This is the function: L(x) = 10log(x) where L is the decibel level and x is the intensity measured in watts per square meter. First calculate the decibel level for each sound. Then, plot the values on the graph, where the y-axis represents decibel level. Sound Softest noise that can be heard Whisper across room Intensity (measured in watts/m 2 ) Decibel Level Light rainfall Normal conversation Car driving, heard from 50 ft away Noisy factory Subway train 0.01 Rock concert 0.1 Shotgun blast 100

8 You have just practiced finding decibel levels given the intensity. Now, use the same function (but solving in the other direction) to find the intensity of the following sounds: Hockey Crowd: 120 decibels Intensity: Heavy City Traffic: 90 decibels Intensity: Dripping Faucet: 25 decibels Intensity: Threshold for Human Pain: 130 decibels Intensity: Now, let s compare intensities of sounds. Suppose that Kade said I love logarithms!, and his voice was measured at 40 decibels. Then, Lorenzo shouted, People, I finished my homework!, and his voice was measured to be 80 decibels. How many times more intense was Lorenzo s voice? To do this, first find the intensity of each sound, just like you were doing above. Kade s intensity: Lorenzo s intensity: Now, find the ratio of Lorenzo s intensity to Kade s intensity and simplify: Notice that Kade was twice as loud (80 is twice as loud as 40), but his intensity was times greater! And now, on to the review packet! Next class is on May 2 nd (B-Day) or May 3 rd (A-Day). You will be finishing your projects. The unit test will be May 6 th (B-Day) or May 7 th (A-Day).

9 Pre-AP Algebra 2 Review Packet Name: The Big Logarithm Review Packet! Unit test is coming next week do a couple pages a day! Part 1: Rational Exponents The definitions: a 1/ n n a a m/ n n a m Evaluate each expression. Converting back and forth between radical and exponential form is helpful. Changing decimals to fraction form is also helpful. 1) (125) 2/3 2) 49 3/2 3) (27) 5/3 4) (64) 1/3 5) 100 3/2 6) /3 7) 64 3/2 8) /2 9) /3 10) ) ) ) ) ) 11 13) ) ) Simplify each expression by using the properties of exponents. 19) 8 5/3 8 1/3 20) x 2/3 1/ 2 = 21) 53/ 4 121/5 22) 1/ /5 23) 35 1/5 35 3/5 5/4 24) / 2 Name Product of Powers Power of a Power Power of a Product Quotient of Powers Power of a Quotient Property a m a n a mn a m n a mn ab n a n b n a m a n a b am n n an b n

10 Part 2: Graphing Exponential Functions Determine if each function will show exponential growth or decay. Also determine the y-intercept. Then make a table of values and graph each function on the coordinate plane. 1) y x 2) y x 3) y 4(2) x 4) y 2(1.5) x

11 Part 3: Percents Convert each percent to a fraction, and reduce. 1) 5% 2) 24% 3) 140% 4) 22.5% 5) 800% Find each percent. 6) 82% of 154 7) 15.4% of 244 8) 2.5% of 15 9) 100% of ) 200% of ) 1000% of 75 Answer each question, thinking about the best way to calculate percent increase/decrease. 12) At a restaurant, you are supposed to leave a tip that is at least 15% of the bill. Your bill was $45. How much do you leave in total? 13) Your beautiful new car was worth $18,000, until the side got dented in. Now, it has lost 23% of its original value. How much is the car worth? 14) You invested $6000 into your savings account. A few years later, the amount grew to $9500. What percent did your savings increase by? 15) I really like to eat tofu, so I buy it in bulk. I had a 12-pound box that I bought a few weeks ago; I just weighed it again, and now there are only 7.5 pounds left. What percent did my total amount of tofu decrease by? Part 4: Exponential Modeling and the Number e For each situation, determine a) the initial value; b) if it is growth or decay; c) the percent increase/decrease. 1) The value of a car over time is modeled by the function v(t) = 35,000(0.66) t, where v is in dollars and t is in years. 2) The value of my sister s house in San Jose is modeled by v(t) = 450,000(1.09) t, where v is in dollars and t is in years.

12 3) A waste product of producing nuclear energy is radioactive plutonium. A given quantity of plutonium is stored in a container. The amount P (in grams) of plutonium present in the container after t years can be modeled by P = 1000(0.993) t. For each problem, write a function that models the situation. Then, answer the questions. 1) The petting zoo director purchased 10 guinea pigs. Each month, the number of guinea pigs increase by 85%. a. Function: b. How many guinea pigs will there be after 8 months? c. How long will it take for there to be 100,000 guinea pigs? (That s a lot!) 2) When you drop a sugar cube into a glass of hot water, the amount of sugar remaining in the cube decreases by 10% per second. Assume that a sugar cube weighs 3 grams. a. Function: b. How many grams will there be left after 10 seconds? c. How long will it take for there to be only ½ gram left? 3) You put $3000 into a savings account that has continuously compounded interest, at an annual rate of 7.5%. a. Function: b. How much money will you have after 3 years? c. How long will it take for your money to triple? 4) A scientist has recently discovered 500 grams of radioactive material at a local park. Each week, she measures less material from the week before and she is able to come up with the following model for the decay: A(t) = 500e -.035t where A is the total grams of radioactive material and t is the number of weeks. a. How many grams will be left after 4 weeks? b. How long will it take for there to be only ¼ of the original amount left?

13 Part 5: Estimating and Evaluating Logarithms To convert between log and exponential form: log b x a b a x Estimate the value of each logarithm (find two consecutive integers that it falls between) without using your calculator. 1) log ) log ) log ) log 1/2 (1/10) Evaluate each logarithm without using your calculator. 5) log ) log 1/4 16 7) log 7 (1/343) 8)log 1/3 (1/81) 9) log ) ln e 11) log ) log Find the value of x in each log expression. 13) log x 32 = 5 14) log 3 x = -6 15) log 1/2 8 = x 16) log x 7 = ½ 17) log x = 7 18) ln e 5 = x 19) log.01 = x 20) log 8 x = 1/3 Change each logarithm to base 10 and then use your calculator s LOG button to estimate its value. 21) log ) log ) log 2/3 (9/11) Change each logarithm to base e and then use your calculator s LN button to estimate its value. 24) log ) log ) log 2/3 (9/11) Solve each equation by isolating the power, then converting to a logarithm. Write the exact answer (as a logarithm), and then use the Change of Base Theorem and your calculator to estimate the solution to the thousandths place. 27) 7 x = 256 Exact: Estimate: 28) 5.2(3.1 x ) = Exact: Estimate: 29) -2(2.5 x ) + 39 = 3 Exact: Estimate:

14 Part 6: Properties of Logarithms Special Cases (Matching Bases): b log b M and log b b M both equal M. The three properties: Multiplication: log b AC log b A log b C Division: A log b C log A log C b b Power: log b A C C log b A Use log t A 0.4 and log t B 2 to find the exact value of each expression: 1) log t AB 2) log t B A 3) log t A 3 4) log t A 3 B 5) log t AB 2 B 6) log t A Expand each expression using the properties of logs. 7) log 2 8x 3 5 8) log 7 x 9) log 7 49x 5 y ) ln x 2 11) log2x 3 y Condense each expression using the properties of logs. 12) log 5 3 log ) 2log x 3log2 14) 1 ln x 3ln5 2 15) 2log 5 8 log log 3 16) 2log x 3log log27 17) log 5 (x 1) log 5 (x 3) 18) log2x 5x 2 3log x

15 Part 7: Solving Logarithmic and Exponential Equations Solve each equation. Make sure to check your answers and cross out any extraneous solutions. 1) 4 log 5 x log ) 1 2 log 5 x 2log 5 3 3) log 2 (x 1) log 2 (x 2) 2 4) log x log(x 5) 2 For 5 and 6, solve by isolating the power and then taking log 10 of both sides. 5) 2(3.5) x ) 2 3 (0.25)x ) 2 x5 3 14x 8) 10 x2 2 x1

16 HW 9-8 Answer Key 1) x = 38 2) x = ±2 3) x = 11 4) x = 4 or x = -1 5) x = 14 6) x = (ln4)/ ) x x log ) 9) x 2, x 1 10) x log log , same 11) 044

17 HW 9-8 Tally Sheet 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) HW 9-8 Tally Sheet 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)