Exponential and Logarithmic Models Section 3.5 Precalculus PreAP/Dual, Revised ©2018 viet.dang@humbleisd.net 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

Compound Interest Equation 𝑨 = Total Amount Earned 𝑷 = Principle 𝒓 = Interest Rate 𝒏 = Compounded Amount 𝒕 = Time 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Video 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 1 $𝟓,𝟎𝟎𝟎 is deposited in an account that pays 𝟔% annual interest compounded quarterly. Find the balance after 25 years. 𝑨= ? 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 1 $𝟓,𝟎𝟎𝟎 is deposited in an account that pays 𝟔% annual interest compounded quarterly. Find the balance after 25 years. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

Compounded Continuously 𝑨 = Total Amount Earned 𝑷 = Principle 𝒆 = The Natural Base 𝒓 = Interest Rate 𝒕 = Time 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 5 An investment of $3,500 at 3% annual interest compounded continuously was made. How much is in the account after 4 years? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

Exponential Growth/Decay 𝑷 = Ending Amount 𝑷 𝟎 = Initial Amount 𝒆 = The Natural Base 𝒌 = Growth or Decay Rate 𝒕 = Time 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 7 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 8 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 9 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 9 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 10 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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 0.0170. Round to the nearest whole number. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 11 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 11 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? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 11 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 = ?? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 12 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 12 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 = ?? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Your Turn The half-life of carbon-14 is 5,730 years. If it is determined that an old bone contains 𝟖𝟓% of its original carbon-14, how old is the bone? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

Newton’s Law of Cooling 𝑻 𝑭 = Final Temperature 𝑻 𝑹 = Temperature of the Environment 𝑻 𝟎 = Initial Temperature of the Object 𝒆 = The Natural Base 𝒌 = Growth or Decay Rate 𝒕 = Time 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 13 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 𝟏𝟐.𝟑𝟓 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 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models 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. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 14 When an object is removed from a furnace and placed in an environment with a constant decay rate of 𝟎.𝟑𝟏𝟏𝟒 and the room temperature of 𝟖𝟎°𝑭, its core temperature is 𝟏𝟓𝟎𝟎°𝑭. If the final temperature is at 𝟑𝟕𝟖°𝑭, about how long is it out of 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 = 𝟎.𝟑𝟏𝟏𝟒 𝒕 = ?? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 14 When an object is removed from a furnace and placed in an environment with a constant decay rate of 𝟎.𝟑𝟏𝟏𝟒 and the room temperature of 𝟖𝟎°𝑭, its core temperature is 𝟏𝟓𝟎𝟎°𝑭. If the final temperature is at 𝟑𝟕𝟖°𝑭, about how long is it out of the furnace (in hours)? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Example 14 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 if is 𝟖𝟎°𝑭 outside, what is the decay factor? Round to three decimal places. 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Your Turn Devin baked a yam at 𝟑𝟓𝟎°, and when Devin removed it from the oven, he let the yam cool, which has a room temperature of 𝟔𝟖°𝑭. After 𝟏𝟎 minutes, the yam has cooled to 𝟐𝟒𝟎°𝑭. What is the decay factor? 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models

§3.5: Exponential and Logarithmic Models Assignment Worksheet 2/27/2019 6:28 AM §3.5: Exponential and Logarithmic Models