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Chapter 8-x Applications of Exponential and Logarithmic Functions Day 1

Chapter 8-x Applications of Exponential and Logarithmic Functions Day 1. Essential Question: What are some types of real-life problems where exponential or logarithmic equations can be used?. 8-x: Exponential/Logarithmic Applications. Compound Interest A = P(1 + ) nt , where:

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Chapter 8-x Applications of Exponential and Logarithmic Functions Day 1

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  1. Chapter 8-xApplications of Exponential and Logarithmic FunctionsDay 1 Essential Question: What are some types of real-life problems where exponential or logarithmic equations can be used?

  2. 8-x: Exponential/Logarithmic Applications • Compound Interest • A = P(1 + )nt, where: • A = Amount at the end of compounding • P = Principal (starting) amount • r = Interest rate (as a decimal) • n = Number of times per year compounded • t = number of years • We’re only going to be dealing with situations where interest is compounded yearly, so we will use the formula: • A = P(1 + r)t

  3. 8-x: Exponential/Logarithmic Applications • Example • A computer valued at $6500 depreciates at the rate of 14.3% per year. • Write a function that models the value of the computer. • A = P(1 + r)t • Do we know A (the value at the end)? • Do we know P (the value at the start)? • Do we know r (the rate)? • Do we know t (the number of years)? • Find the value of the computer after three years. depreciates No Yes, 6500 Yes, 0.143 No A = 6500(1 – 0.143)t = 6500(0.857)t A = 6500(0.857)3 = $4091.25

  4. 8-x: Exponential/Logarithmic Applications • Continuously Compounding Interest • So let’s bring “n” back for just one example: • Suppose you invest $1 for one year at 100% annual interest, compounded n times per year. Find the maximum value of the investment in one year. • Observe what happens to the final amount as n grows larger and larger.

  5. 8-x: Exponential/Logarithmic Applications • Compounding Continuously • Annually • Semiannually • Quarterly • Monthly • Daily • Hourly: 365 • 24 = 8760 periods • Every minute: 8760 • 60 = 525,600 periods • Every second: 525,600 • 60 = 31,536,000 periods

  6. 8-x: Exponential/Logarithmic Applications • $2.7182825 is the same as the number e to five decimal places (e = 2.71828182…) • So if we’re compounding continuously (instead of yearly), we can instead use the equation • A = Pert, where • A = Amount at the end of compounding • P = Principal (starting) amount • r = Interest rate (as a decimal) • t = number of years

  7. 8-x: Exponential/Logarithmic Applications • Example • Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. How much money will you have in the account after 5 years? • A = Pert • Do we know A (the value at the end)? • Do we know P (the value at the start)? • Do we know r (the rate)? • Do we know t (the number of years)? No Yes, 1050 Yes, 0.055 Yes, 5 A = 1050 e(0.055•5) = $1382.36

  8. 8-x: Exponential/Logarithmic Applications • Your Turn • Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. How much money will you have in the account after 3 years? • A = Pert A = 1300(e)(0.043•3) = $1479.00

  9. 8-x: Exponential/Logarithmic Applications • Assignment • Worksheet • Round all problems appropriately (if talking about money: 2 decimal places; if talking about population: nearest integer)

  10. Chapter 8-xApplications of Exponential and Logarithmic FunctionsDay 2 Essential Question: What are some types of real-life problems where exponential or logarithmic equations can be used?

  11. 8-x: Exponential/Logarithmic Applications • Radioactive Decay • The half-life of a radioactive substance is the time it takes for half of the material to decay. • It’s most often used for things like carbon-14 dating, which determines how old a substance is • The function for radioactive decay is where: • P = the initial amount of the substance • x = 0 corresponds to the time since decay began • h = the half-life of the substance

  12. 8-x: Exponential/Logarithmic Applications • Example • A hospital prepares a 100-mg supply of technetium-99m, whichhas a half-life of 6 hours. Write an exponential function to find the amount of technetium-99m. • Do we know y (the value at the end)? • Do we know P (the value at the start)? • Do we know x (the amount of time)? • Do we know h (the half-life)? • Use your function to determine how much technetium-99m remains after 75 hours. No Yes, 100 No Yes, 6

  13. 8-x: Exponential/Logarithmic Applications • Example • Arsenic-74 is used to locate brain tumors. It has a half-life of 17.5 days. Write an exponential decay function for a 90-mg sample. • Use the function to find the amount remaining after 6 days.

  14. 8-x: Exponential/Logarithmic Applications • Loudness • Logarithms are used to model sound. The intensity of a sound is a measure of the energy carried by the sound wave. The greater the intensity of a sound, the louder it seems. This apparent loudness L is measured in decibels. You can use the formula, where • I is the intensity of the sound in watts per square meter (W/m2) • I0is the lowest-intensity sound that the average human ear can detect. (We will use I0 = 10-12 w/m2)

  15. 8-x: Exponential/Logarithmic Applications • Example • Suppose you are the supervisor on a road construction job. Your team is blasting rock to make way for a roadbed. One explosion has an intensity of 1.65 x 10-2 W/m2. What is the loudness of the sound in decibels?

  16. 8-x: Exponential/Logarithmic Applications • Assignment • Worksheet • Round all problems appropriately • For half life, round to two decimal places • For loudness, round to the nearest dB

  17. Chapter 8-xApplications of Exponential and Logarithmic FunctionsDay 3 Essential Question: What are some types of real-life problems where exponential or logarithmic equations can be used?

  18. 8-x: Exponential/Logarithmic Applications • Acidity • Scientists use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance. The pH of a substance equals:pH = –log[H+],where [H+] is the concentration of hydrogen ions.

  19. 8-x: Exponential/Logarithmic Applications • Example • The pH of lemon juice is 2.3, while the pH of milk is 6.6. Find the concentration of hydrogen ions in each substance. Which substance is more acidic? • Lemon Juice Milk (Your turn) • pH = -log[H+]2.3 = -log[H+] substitutelog[H+] = -2.3 divide by -1[H+] = 10-2.3convert to exp[H+] = 5.0 x 10-3 pH = -log[H+] 6.6 = -log[H+] log[H+] = -6.6 [H+] = 10-6.6 [H+] = 2.5 x 10-7 The lemon juice is more acidic as it contains more hydrogen ions

  20. 8-x: Exponential/Logarithmic Applications • Compound Interest (finding something other than A) • Remember our formulas from Monday:A = P(1 + r)t for compounding annually A = Pert for compounding continuously • Finding P • How much should be invested at 5.5% compounded annually to yield $3500 at the end of 4 years? • 3500 = P(1 + 0.055)4 3500 = P(1.055)4 3500/(1.055)4 = P $2825.26= P

  21. 8-x: Exponential/Logarithmic Applications • Compound Interest (finding something other than A) • Finding r • What interest rate would be required to grow an investment of $1000 to $1407.10 in seven years if that interest is compounded annually? • 1407.10 = 1000(1 + r)7 1.4071 = (1 + r)7 1.05 = 1 + r 0.05 = r, meaning an interest rate of 5%

  22. 8-x: Exponential/Logarithmic Applications • Compound Interest (finding something other than A) • Finding t • How long will it take to double an investment of $500 at 7% interest, compounded annually? • 1000 = 500(1 + 0.07)t 2 = (1.07)t log1.07 2 = t log 2/log 1.07 = t 10.25 = t, meaning 10.25 years

  23. 8-x: Exponential/Logarithmic Applications • Assignment • Worksheet • Round all problems appropriately • If talking about money: 2 decimal places

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