7.5b - Applications of Logarithms

7.5b - Applications of Logarithms
Worksheet Key 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Applications of Logarithms
Section 7.5B, Revised ©2013, 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Compound Interest Equation
A = Total Amount Earned P = Principle r = Interest Rate n = Compounded Amount t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Compound Interest Equation
Video 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Compounded Time Frames
Annually: 1 time a year Semi-Annually: 2 times a year Quarterly: 4 times a year (not THREE TIMES a year) Monthly: 12 times a year Daily: 365 times a year 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 1 $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years. A = ? Do we know how much it is when the balance after 25 years? P = $5,000 $5,000 is deposited r = 0.06 Interest Rate – remember it needs to be in decimal form n = 4 Compounded quarterly t = 25 Time it takes to accrue amount 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 1 $5,000 is deposited in an account that pays 6% annual interest compounded quarterly. Find the balance after 25 years. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 2 Determine the amount that a $5,000 investment over ten years at an annual interest rate of 4.8% is worth compounded daily. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 3 How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn A deposit is made for $100,000 into an account that pays 6% interest. Find the balance after 10 years if the interest is compounded monthly. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Compound Continuously Formula
A = Total Amount Earned P = Principle e = The Natural Base r = Interest Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Review A deposit is made for $100,000 into an account that pays 6% interest. Find the balance after 10 years if the interest is compounded quarterly. A = ?? P = $100,000 r = 0.06 n = 4 t = 10 A = Amount P = Principle r = Interest Rate n = Compounded t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 4 A deposit is made for $100,000 into an account that pays 6% interest. Find the balance after 10 years if the interest is compounded continuously. A = ?? P = $100,000 e = Use e in Calc r = 0.06 t = 10 A = Account P = Principle e = Natural Base r = Interest Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 5 An investment of $3,500 at 3% annual interest compounded continuously was made. How much is in the account after 4 years? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn Suppose that you put in $1,000 into a savings account that compounded continuously. Determine the amount with an interest rate of 5.1% after 10 years. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 6 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? Doubled Amount 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 6 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 7 You have deposited $2,500 in an account that pays 8.5% interest, compounded continuously. How long will it take your money to triple? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Isolate the variable Condense the problem Convert the equation, if necessary 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Exponential Growth/Decay
P = Ending Amount P0 = Initial Amount e = The Natural Base k = Growth or Decay Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 8 A certain bacterium has an exponential growth rate of 25% per day. If we start with 0.5 grams and provide unlimited resources how much bacteria can we grow in 14 days? P = ?? P0 = 0.5 e = The Natural Base k = 0.25 t = 14 days P = Ending Amount P0 = Initial Amount e = The Natural Base k = Growth or Decay Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 9 What is the total amount of bacteria when the initial amount of bacteria is 300, k = 0.068, and the time studied is 52 hours? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02, what will be the population after 5 hours? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 10 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate? P = 12,000 P0 = 5,000 e = The Natural Base k = ?? t = 10 P = Ending Amount P0= Initial Amount e = The Natural Base k = Growth or Decay Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 10 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 11 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 15,000 cells in 12 hours. What is the growth rate? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn The population of a certain city in 2000 was 99,500. What is its initial population in 1975 when its growth rate is at Round to the nearest whole number. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 12 If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left? P = 75 P0 = 150 e = The Natural Base k = ?? t = 4.2 P = Ending Amount P0= Initial Amount e = The Natural Base k = Growth or Decay Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 12 If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 12 If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left? P = 10 P0 = 150 e = The Natural Base k = –.1650 t = ?? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 13 The half-life of carbon-14 is 5,730 years. The skeleton of a mastodon has 42% of its original Carbon-14. When did the mastodon die? P = ½ (half life) P0 = 1 (full life) e = The Natural Base k = ?? t = 5,730 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 13 The half-life of carbon-14 is 5,730 years. The skeleton of a mastodon has 42% of its original Carbon-14. When did the mastodon die? P = 0.42 (total left) P0 = 1 e = The Natural Base k = (ln 0.5)/5730 t = ?? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn The half-life of carbon-14 is 5730 years. If it is determined that an old bone contains 85% of its original carbon-14, how old is the bone? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Newton’s Law of Cooling
TF = Final Temperature TR = Temperature of the Environment T0 = Initial Temperature of the Object e = The Natural Base k = Growth or Decay Rate t = Time 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 14 A container of ice cream arrives home from the supermarket at a temperature of 65°F. It is placed in the freezer which has a temperature of 20°F. Determine the final temperature at which it will be still considered “freezing,” if the rate of change is 0.107°F per minute for minutes. TF = Final Temperature TR = Environment Temp T0 = Initial Temperature e = The Natural Base k = Growth or Decay Rate t = Time TF = ?? TR = 20° T0 = 65° e = The Natural Base k = 0.107 t = 12.35 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn The cooling model for tea served in a 6 oz. cup uses Newton’s Law of Cooling equation. The original temperature was 200°F and current environment temperature of the tea is at 68°F. Determine the temperature if the decay rate is at 0.41 per minute and waiting time is 6 minutes. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 15 When an object is removed from a furnace and placed in an environment with a constant decay rate of and an environmental temperature of 80F, its core temperature is 1500°F. If the core temperature is at 378°F, about how long is it in the furnace (in hours)? TF = Final Temperature TR = Environment Temp T0 = Initial Temperature e = The Natural Base k = Growth or Decay Rate t = Time TF = 378° TR = 80° T0 = 1500° e = The Natural Base k = t = ?? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 15 When an object is removed from a furnace and placed in an environment with a constant decay rate of and an environmental temperature of 80F, its core temperature is 1500°F. If the core temperature is at 378°F, about how long is it in the furnace (in hours)? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Example 16 Pete was driving on a hot day when the car starts overheating and stops running. It overheats to 280°F and can be driven again at 230°F. Suppose it takes 60 minutes until Pete can drive it is 80°F outside, what is the decay factor? Round to three decimal places. 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Your Turn Devin baked a yam at 350°, and when Devin removed it from the oven, he let the yam cool, which has a temperature of 68°F. After 10 minutes, the yam has cooled to 240°F. What is the decay factor? 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

Assignment Worksheet 1/1/2019 9:55 AM 7.5b - Applications of Logarithms

7.5b - Applications of Logarithms

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