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For questions 1 – 3, do NOT use exponential regression.

For questions 1 – 3, do NOT use exponential regression. The table below shows the temperature T of a certain liquid after it has been cooled in a freezer for m minutes. Show that the data can be modeled with an exponential function.

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For questions 1 – 3, do NOT use exponential regression.

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  1. For questions 1 – 3, do NOT use exponential regression. • The table below shows the temperature T of a certain liquid after it has • been cooled in a freezer for m minutes. Show that the data can be • modeled with an exponential function. • 2. What is the decay factor per minute? Briefly show work. • Write an exponential model for the data. • The defense budget for the United States in various years is show in the table at the right. Use exponential regression to find a model for this data using years since 1997. Round all values to the nearest thousandth. .671/2  .82 The decay factor per minute is .82 T = 12.02(.82m) 0 2 4 5 7 B = 348.454(1.067t) where B = budget in billions of dollars and t = number of years since 1997

  2. Remember to download from D2L and print a copy of the Final Group Project.

  3. Uranium 239 (U239) is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 10 grams were placed in a container, and the amount remaining was measured at 5-minute intervals and recorded in the table below. U Periodic Table of Elements • Show that an exponential model would be appropriate for this data. • What is the decay factor per minute (nearest thousandth)? • Find an exponential model for the data. • What is the decay rate each minute? What does this number mean in practical terms? • Use functional notation to express the amount remaining after 13 minutes and then calculate that value. • The half-life of a radioactive element is the time it takes for the mass to decay by half. Use the graphing calculator to find the half-life of U239 to the nearest hundredth of a minute. G(t) = 10

  4. Uranium 239 (U239) is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 10 grams were placed in a container, and the amount remaining was measured at 5-minute intervals and recorded in the table below. U Periodic Table of Elements 6. The half-life of a radioactive element is the time it takes for the mass to decay by half. Use the graphing calculator to find the half-life of U239 to the nearest hundredth of a minute. G(t) = 10 23.55 minutes

  5. x = 23.55 Our goal for the rest of this class period is to find an algebraic way of solving the equation 5 = 10(.971x) (and others like it)

  6. Answers to even-numbered HW problems Section 4.3 S-2 Exponential S-10y = 5.592(.814x) Ex 6 Suspect data point

  7. x = 23.55 10 10 Our goal for the rest of this class period is to find an algebraic way of solving the equation 5 = 10(.971x) (and others like it) .5 = .971x Since the variable is part of the exponent, the only way to isolate the variable is to use logarithms.

  8. 10x = 10

  9. 101 = 10

  10. 101 = 10 10x = 100

  11. 101 = 10 102 = 100

  12. 101 = 10 What is the value of x? 10x = 40 102 = 100

  13. 101 = 10 What is the value of x? 10x = 40 102 = 100 The question can be rephrased: If 10 is raised to a power and the result is 40, what is the exponent?

  14. 101 = 10 10x = 40 102 = 100 The question can be rephrased: If 10 is raised to a power and the result is 40, what is the exponent? The word logarithm is a synonym for exponent.

  15. 101 = 10 10x = 40 102 = 100 The question can be rephrased: If 10 is raised to a power and the result is 40, what is the exponent? This question can be rephrased: If 10 is the base, what is the logarithm that will give a value of 40?

  16. x = log1040 101 = 10  1.6 10x = 40 102 = 100 The question can be rephrased: If 10 is raised to a power and the result is 40, what is the exponent? This question can be rephrased: What is log1040 ? exponent base 10 result

  17. log1040 = 1.6 The word logarithm is a synonym for exponent. The common logarithm of a, written log10a, is defined as the power of 10 that gives a. The common logarithm of 40 is 1.6 because using base 10, the exponent needed to get a value of 40 is 1.6. Can we get a more accurate value for log1040 ? log1040  1.602059991

  18. 5 32 2 Is it true that 25 = 32 ? Identify the exponent, the base, and the result. Write an equivalent statement using logarithms. log232 = 5

  19. 2.68 10 478.63 Is it true that 102.68 = 478.63 (approximately)? Identify the exponent, the base, and the result. Write an equivalent statement using logarithms. log10 478.63 = 2.68 When no base is indicated, it is understood to be 10. log478.63 = 2.68

  20. When the base of a logarithm is 10, we call it a common logarithm, and we do not indicate the base. log1040 is the same as log 40.

  21. Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 723,456 b) log 0.0000245 c) log (4) When the base of a logarithm is 10, we call it a common logarithm, and we do not indicate the base. log1040 is the same as log 40. Write an equivalent exponential equation for each. 105.8594 = 723,456 = 5.8594 104.6108 = .0000245 = 4.6108 ERR: nonreal ans

  22. An important property of logarithms: log(ab) = b (loga) Illustration: Does log(27) = 7(log2)?

  23. An important property of logarithms: log(ab) = b (loga) Illustration: Does log(27) = 7(log2)?

  24. 10 10 x = = 23.52 -.3010 -.0128 Now let’s solve the equation x = 23.55 5 = 10(.971x) Divide both sides by 10 .5 = .971x log(.5) = log(.971x) Take the logarithm on both sides log(.5) = xlog(.971) Apply the property = x (-.0128) -.3010 Compute the values Solve for x

  25. 10 10 Now let’s solve the equation x = 23.55 5 = 10(.971x) Divide both sides by 10 .5 = .971x log(.5) = log(.971x) Take the logarithm on both sides log(.5) = xlog(.971) Apply the property = x Solve for x 23.55 = x

  26. Solve algebraically to the nearest thousandth

  27. Solve algebraically to the nearest thousandth 48 48 15.3 = log(15.3) = log log(15.3) = x log (3.7) log(15.3) = x log (3.7) 2.085 = x

  28. Section 4.4 (Do not read) Handout – Common Logs (correction to question 18. It should read “Solve question 17 algebraically.”)

  29. Additional Practice The table below shows the average weekly amount of electricity used in five Michigan cities in 2011. • 1. Make a scatter plot of the data. Based on the graph, would • an exponential model be appropriate? • Write an equation for an exponential model for the weekly • amount of electricity used versus the population. Round • coefficients to three decimal places (nearest thousandth). • 3. Graph the exponential model. • 4. What is the growth rate in weekly electricity use per 1,000 people? • 5. Use your model to estimate the weekly amount of electricity used in 2011 in a • Michigan city with a population of 90,000 people. • 6. The weekly amount of electricity used by the people of another Michigan city in 2011 • was 660 thousand kilowatts. Use your graph to estimate the population of the city • that year. • 7. Solve question 6 algebraically.

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