Skills Practice Skills Practice for Lesson 8.1

Transcription

1 Skills Practice Skills Practice for Lesson.1 Name Logs, Exponents, and More Solving Exponential and Logarithmic Equations Date Problem Set Solve each logarithmic equation by first converting to an exponential equation. 1. log 3 x 4 2. log x 3.2 log 3 x 4 x 3 4 x 1 log ln x 7 4. log 4 x 2 5. log x 0 6. log 5 x log 2 16 x. log 3 x Chapter l Skills Practice 601

2 9. ln x 10. log x 11. log x 12. log x Solve each logarithmic equation by converting to an exponential equation and applying the properties of exponents. 13. log x log x log x 27 3 x 3 27 x 3 log log x log x Chapter l Skills Practice

3 Name Date 17. log x log x log 256 x 20. log x ln x log x ln x log x Chapter l Skills Practice 603

4 604 Chapter l Skills Practice

5 Skills Practice Skills Practice for Lesson.2 Name Decibels, ph, and the Richter Scale Logarithms and Problem Solving Part I Vocabulary Match each definition to its corresponding term. Date 1. a scale for measuring the acidity or a. decibel alkalinity of a substance 2. a scale used to measure the intensity b. ph scale of earthquakes 3. a unit of measure for the loudness c. Richter scale of sound Problem Set Use the formula for the loudness of a sound, db 10 log I I 0, where db is the decibel level, I 0 is the intensity of the threshold sound, and I is the intensity of other sounds, to solve each problem. 1. The sound of a vacuum cleaner is about 10,000,000 times as intense as the threshold sound, I 10,000,000I 0. What is the decibel level for a vacuum cleaner? db 10 log I I 0 db 10 log 10,000,000I 0 I 0 db 10 log 10,000,000 db 70 A vacuum cleaner has a decibel level of 70. Chapter l Skills Practice 605

6 2. The sound of a whisper is about 100 times as intense as the threshold sound, I 100I 0. What is the decibel level of a whisper? 3. The sound on an aircraft carrier deck when an aircraft launches is about 100,000,000,000,000 times as intense as the threshold sound, I 100,000,000,000,000I 0. What is the decibel level on the aircraft carrier deck? 4. The sound of a chain saw has a decibel level of 110. The sound of the chain saw is how many times more intense than the threshold sound? 606 Chapter l Skills Practice

7 Name Date 5. The sound of the background music while on hold during a telephone call has a decibel level of 60. The background music is how many times more intense than the threshold sound? 6. The sound of a dishwasher has a decibel level of 0. The sound of the dishwasher is how many times more intense than the threshold sound? Use the formula for ph, ph log H, where H is the concentration of hydrogen ions, to solve each problem. 7. The H concentration in seawater is 10. What is the ph level of the seawater? Is seawater acidic or basic? ph log H ph log 10 ph.0 The ph level of seawater is about.0. The ph level is greater than 7, so seawater is basic. Chapter l Skills Practice 607

8 . The concentration of hydrogen ions in black coffee is What is the ph level of the coffee? Is black coffee acidic or basic? 9. The concentration of hydrogen ions in battery acid is 0.1. What is the ph level of the battery acid? Is battery acid acidic or basic? 10. Milk has a ph of 6.6. What is the concentration of hydrogen ions in milk? 60 Chapter l Skills Practice

9 Name Date 11. Ammonia has a ph of What is the concentration of hydrogen ions in ammonia? 12. Tomatoes have a ph of 4.5. What is the concentration of hydrogen ions in tomatoes? Use the formula M log I I 0, where M is the magnitude of an earthquake on the Richter scale, I 0 represents the intensity of a zero-level earthquake the same distance from the epicenter, and I is the number of times more intense an earthquake is than a zero-level earthquake, to solve each problem. 13. An earthquake southwest of Chattanooga, Tennessee in 2003 had a seismographic reading of millimeters registered 100 kilometers from the center. What was the magnitude of the Tennessee earthquake of 2003 on the Richter scale? M log ( I I 0 ) M log ( ) M 4.9 The Tennessee earthquake of 2003 measured 4.9 on the Richter scale. Chapter l Skills Practice 609

10 14. An earthquake in Illinois in 200 had a seismographic reading of 15.5 millimeters registered 100 kilometers from the center. What was the magnitude of the Illinois earthquake of 200 on the Richter scale? 15. An earthquake off the northern coast of California in 2005 had a seismographic reading of 15,49 millimeters registered 100 kilometers from the center. What was the magnitude of the California earthquake in 2005 on the Richter scale? 16. The devastating earthquake in Haiti in 2010 had a magnitude of 7.0 on the Richter scale. What was its seismographic reading in millimeters 100 kilometers from the center? 610 Chapter l Skills Practice

11 Name Date 17. Calculate the value of the seismographic reading for an earthquake of magnitude 6.4 on the Richter scale. 1. Calculate the value of the seismographic reading for an earthquake of magnitude.1 on the Richter scale. Chapter l Skills Practice 611

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13 Skills Practice Skills Practice for Lesson.3 Name Date Depreciation, Population Growth, and Radioactive Decay Logarithms and Problem Solving Part II Problem Set log Use the formula t ( V C ), where t is the age of an item in years, log(1 r ) V is the value of the item after t years, C is the original value of the item, and r is the yearly rate of depreciation expressed as a decimal, to solve each problem. 1. A luxury car was originally purchased for $110,250 and is currently valued at $65,200. The average rate of depreciation for this car is 10.3% per year. How old is the car to the nearest tenth of a year? log t ( V C ) log(1 r ) log ( 65, ,250 ) t log( ) t 4. The car is about 4. years old. 2. A van was originally purchased for $3,950 and is currently valued at $3,600. The average rate of depreciation for this van is 26.9% per year. How old is this van to the nearest tenth of a year? Chapter l Skills Practice 613

14 3. A motorcycle was originally purchased for $21,000 and is currently valued at $15,500. The average rate of depreciation for this motorcycle is 5.% per year. How old is this motorcycle to the nearest tenth of a year? 4. A 4-year old car was originally purchased for $35,210. Its current value is $16,394. What is this car s annual rate of depreciation? 614 Chapter l Skills Practice

15 Name Date 5. A 7-year old car is currently valued at $4,500. The car depreciates at the rate of 24.2% per year. What was the original price of this car? 6. A 5-year old car was originally purchased for $16,900. The car depreciates at the rate of 26.7% per year. What is the current value of this car? Chapter l Skills Practice 615

16 Use the given formula to solve each problem. 7. The formula for the population of a species is n k log(a), where n represents the population of a species, A is the area of the region in which the species lives, and k is a constant that is determined by field studies. Based on population samples, an area that is 1000 square miles contains 360 wolves. Calculate the value of k. Then use the formula to find the number of wolves remaining in 15 years if only 300 square miles of this area is still inhabitable. n k log(a) 360 k log 1000 k 120 The value of k is 120. n 120 log(a) 120 log In 15 years, there will be approximately 297 wolves remaining in the area.. The formula for the population of a species is n k log(a), where n represents the population of a species, A is the area of the region in which the species lives, and k is a constant that is determined by field studies. Based on population samples, a rainforest that is 100 square miles contains 342 monkeys. Calculate the value of k. Then use the formula to find the number of monkeys remaining in 5 years if only 40 square miles of the rainforest survives due to the current level of deforestation. 616 Chapter l Skills Practice

17 Name Date 9. The formula y a b ln t, where t represents the time in hours, y represents the amount of fresh water produced in t hours, a represents the amount of fresh water produced in one hour, and b is the rate of production, models the amount of fresh water produced from salt water during a desalinization process. In one desalination plant, cubic yards of fresh water can be produced in one hour with a rate of production of How much fresh water can be produced after hours? 10. The formula y a b ln t, where t represents the time in hours, y represents the amount of fresh water produced in t hours, a represents the amount of fresh water produced in one hour, and b is the rate of production, models the amount of fresh water produced from salt water during a desalinization process. At a desalination plant, 1.65 cubic yards of fresh water can be produced in one hour with a rate of production of How long will it take for the plant to produce 250 cubic yards of fresh water? Chapter l Skills Practice 617

18 11. The amount of medicine left in a patient s body can be predicted by the formula log t ( C A ), where t is the time in hours since the medicine was administered, log (1 r) C is the current amount of the medicine left in the patient s body in milligrams, A is the original dose of the medicine in milligrams, and r is the rate at which the medicine leaves the body. A patient is given 25 milligrams of a medicine which leaves the body at the rate of 15% per hour. How long will it take until only 4 milligrams of the medicine remain in the patient s body? 12. The amount of medicine left in a patient s body can be predicted by the formula log t ( C A ), where t is the time in hours since the medicine was administered, log (1 r ) C is the current amount of the medicine left in the patient s body in milligrams, A is the original dose of the medicine in milligrams, and r is the rate at which the medicine leaves the body. Ten hours after administering 50 milligrams of the medicine, 10 milligrams remain in the patient s body. At what rate is the medicine leaving the body? 61 Chapter l Skills Practice

19 Name Date The amount of a radioactive isotope remaining can be modeled using the formula A A 0 e kt, where t represents the time in years, A represents the amount of the isotope remaining in grams after t years, A 0 represents the original amount of the isotope in grams, and k is the decay constant. Use this formula to solve each problem. 13. Strontium-90 is a radioactive isotope with a half-life of about 29 years. Calculate the decay constant for Strontium-90. Then find the amount of 100 grams of Strontium-90 remaining after 120 years. A A 0 e kt 1 2 A 0 A 0 e k(29) 1 e 29k 2 ln 1 29k 2 k The decay constant for Strontium-90 is about A 100e 0.024(120) A 5.61 After 120 years, there would be about 5.61 grams remaining. 14. Radium-226 is a radioactive isotope with a half-life of about 1622 years. Calculate the decay constant for Radium-226. Then find the amount of 20 grams of Radium-226 remaining after 500 years. Chapter l Skills Practice 619

20 15. Carbon-14 is a radioactive isotope with a half-life of about 5730 years. Calculate the decay constant for Carbon-14. Then find the amount of 6 grams of Carbon-14 that will remain after 22,000 years. 16. Cesium-137 is a radioactive isotope with a half-life of about 30 years. Calculate the decay constant for Cesium-137. Then calculate the percentage of a Cesium-137 sample remaining after 100 years. 620 Chapter l Skills Practice

21 Name Date 17. Uranium-232 is a radioactive isotope with a half-life of about 69 years. Calculate the decay constant for Uranium-232. Then calculate the percentage of a Uranium-232 sample remaining after 200 years. 1. Rubidium-7 is a radioactive isotope with a half-life of about years. Calculate the decay constant for Rubidium-7. Then calculate the percentage of a Rubidium-7 sample remaining after 1,000,000 years. Chapter l Skills Practice 621

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23 Skills Practice Skills Practice for Lesson.4 Name Money, Money, Money! Loans and Investments Date Problem Set The monthly payment for a car loan or a home mortgage can be determined IP using the formula p 0, where I is the interest rate per payment n 1 (1 I ) period, P 0 is the original value of the loan, p is the monthly payment, and n is the number of payments. The number of monthly payments, n, can be log calculated using the formula n ( 1 IP 0 p ). Use the appropriate formula log(1 I ) to solve each problem. 1. Calculate the monthly payment for a four-year car loan of $40,000 at 3.99%. Then calculate the total amount paid. p p IP 0 1 (1 I ) n ( ) (40,000) 1 ( ) 4 p T $43, The payment would be $902.9 per month for a final total payment of $43, Chapter l Skills Practice 623

24 2. Calculate the monthly payment for a six-year car loan of $26,500 at 5.5%. Then calculate the total amount paid. 3. Calculate the monthly payment on a five-year car loan of $25,000 at.2%. Then calculate the total amount paid. 624 Chapter l Skills Practice

25 Name Date 4. Calculate the number of monthly payments of $500 needed to repay a $10,250 loan at 7.5%. Then calculate the amount of the last payment and the total amount paid. 5. Calculate the number of monthly payments of $325 needed to repay a $5000 loan at.9%. Then calculate the amount of the last payment and the total amount paid. Chapter l Skills Practice 625

26 6. Calculate the number of monthly payments of $50 needed to repay a $2000 loan at 12.9%. Then calculate the amount of the last payment and the total amount paid. 7. Calculate the monthly payment and total amount you would pay for a 30-year mortgage loan of $225,000 at 4.25%. 626 Chapter l Skills Practice

27 Name Date. Calculate the monthly payment and total amount that you would pay for a 25-year mortgage loan of $150,000 at 3.4%. 9. Calculate the monthly payment and total amount of interest you would pay for a 30-year mortgage loan of $399,000 at 5%. Chapter l Skills Practice 627

28 10. Calculate the number of payments of $540 per month that would need to be made on a $79,000 mortgage loan at 4.2%. Then calculate the amount of the last payment and the total amount paid. 11. Calculate the number of payments of $1450 per month that would need to be made on a $19,000 mortgage loan at 6.0%. Then calculate the amount of the last payment and the total amount paid. 62 Chapter l Skills Practice

29 Name Date 12. Calculate the number of payments of $1299 per month you would make on a $279,000 mortgage loan at 3.75%. Then calculate the amount of the last payment and the total amount paid. The amount of money in a savings account can be calculated using the formula P p I [(1 I )n 1], where P is the amount of money in the account, p is the amount that you deposit each month, n is the number of months, and I is the interest per payment period. The formula for the number of log ( 1 IF v p ) months to reach an investment goal is n log(1 I ), where F is the v future value or investment goal, p is the amount that you deposit each month, n is the number of months, and I is the interest per payment period. Use the appropriate formula to solve each problem. 13. How much money would you have saved after years if you put $20 into a savings account each month that pays 2.5% annually? P p I (1 I )n 1 20 P P ( ) 96 1 You would have $ Chapter l Skills Practice 629

30 14. How much money would you have saved after 1 years if you put $50 into a savings account each month that pays 1.75% annually? 15. How many monthly deposits of $500 would need to be made into an account that earns 3.0% interest in order to save $120,000? How many years is this? 16. How many monthly deposits of $100 would need to be made into an account that earns 1.15% interest in order to save $36,000? How many years is this? 630 Chapter l Skills Practice

31 Name Date 17. What monthly deposit would need to be made into an account earning 2.46% interest in order to save $500,000 in 25 years? Chapter l Skills Practice 631

32 1. What monthly deposit would need to be made into an account earning 1.95% monthly interest in order to save $46,000 in 15 years? 632 Chapter l Skills Practice