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Click on the title below you wish to view At the end of the presentation you will find a link to return to the Table of Contents I. Compound Interest Formula II. Continuous Compound Interest Formula III. Population Growth Formula IV. Doubling-Time Growth Formula V. Exponential Decay VI. Half-Life Decay Formula VII. Newton’s Law of Cooling r A P 1 n nt r A P 1 n nt r A P 1 n P nt is the Principle invested The amount of money invested in a Savings Institution The value of an object: House New Car Piece of Property Bicycle r A P 1 n A nt is the total value of the investment after t years r A P 1 n t nt is the number of years the Principle has been invested r A P 1 n r nt is the Rate of interest Expressed as a decimal r A P 1 n nt n is the Number of times the interest is Compounded per year “Compounded annually” n=1 r A P 1 n nt n is the Number of times the interest is Compounded per year “Compounded semi-annually” n=2 r A P 1 n nt n is the Number of times the interest is Compounded per year “Compounded quarterly” n=4 r A P 1 n nt n is the Number of times the interest is Compounded per year “Compounded monthly” n = 12 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 n 1 r A P 1 n nt One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 n 1 r A P 1 n nt A the unknown One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 n 1 r A P 1 n nt A the unknown One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 n 1 r A P 1 n nt A the unknown One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. P 1,000 r 0.12 n 1 t2 r A P 1 n nt A the unknown P 1,000 r 0.12 n 1 t2 r A P 1 n nt A the unknown 0.12 A 1,000 1 1 P 1,000 r 0.12 n 1 12 t2 0.12 A 1,000 1 1 12 A 1,000 1 0.12 A 1,000 1.12 2 2 0.12 A 1,000 1 1 12 A 1,000 1 0.12 A 1,000 1.12 A 1254.4 2 2 A 1254.4 One thousand dollars is invested at 12% interest compounded annually. Determine how much the investment is worth after 2 years. The investment is worth $1,254.40 r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P? r ? n? t ? A? r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P? r ? n? t ? A? r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P 500 r ? r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P 500 r 0.10 n? r A P 1 n nt The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P 500 r 0.10 n 1 t ? r A P 1 n nt A the unknown The value of a new $500 television Decreases 10% per year. Find its value after 5 years. P 500 r 0.10 n 1 t 5 r A P 1 n nt A the unknown P 500 r 0.10 n 1 t 5 r A P 1 n nt A the unknown 0.10 A 500 1 1 P 500 r 0.10 n 1 15 t 5 0.10 A 500 1 1 A 500 1 0.10 5 A 500 0.90 5 15 0.10 A 500 1 1 15 A 500 1 0.10 5 A 500 0.90 5 A 295.25 The television is worth $295.25 r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P? r ? n? t ? A? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P? r ? n? t ? A? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P 100 r ? n? t ? A? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P 100 r .072 n? t ? A? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P 100 r .072 n4 t ? A? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P 100 r .072 n4 t unknown A ? r A P 1 n nt One hundred dollars is invested at 7.2% interest compounded quarterly. How long will it take for the investment to double? P 100 r .072 n4 t unknown A 200 r A P 1 n P 100 r .072 n4 nt t unknown A 200 r A P 1 n nt .072 200 100 1 4 P 100 r .072 n4 4t t unknown A 200 .072 200 100 1 4 200 100 1 .018 200 100 1.018 4t 4t 4t 200 100 1.018 4t 200 100 1.018 4t 200 100 1.018 100 4t 100 2 1.018 4t log 2 log 1.018 4t log 2 4t log 1.018 log 2 4t log 1.018 4log 1.018 4log 1.018 log 2 t 4log 1.018 log 2 4t log 1.018 4log 1.018 4log 1.018 log 2 t 4log 1.018 9.7 t It takes approximately 10 years Nearest Year Link Back to Table of Contents A Pe rt Where did this formula come from? r A P 1 n nt If Compounded Continually Then how many times per year? Then n is approaching infinity r A P 1 n n Let x r nt n approaches infinity r remains constant x approaches infinity r A P 1 n n If r x r r nt n xr r A P 1 n n If r x r r xrt n xr r A P 1 n n If x r xrt 1 r x n 1 A P 1 x n If x r xrt 1 r x n 1 A P 1 x xrt 1 1 x 1 A P 1 x xrt x x approaches infinity 1 1 x x 1 Y1 1 x nd 2 TABLE x approaches infinity x 1 1 x x 1 1x x x approaches infinity 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183 2.7183 x 1 1 x x 1 1x 4 x 2.4414 8 x 12 x approaches infinity 100 365 10,000 100,000 1,000,000 1 1 x x 1 1x 4 8 2.4414 2.5658 x x 12 x approaches infinity 100 365 10,000 100,000 1,000,000 1 1 x x 1 1x x 4 8 12 2.4414 2.5658 2.6130 x x approaches infinity 100 365 10,000 100,000 1,000,000 1 1 x x 1 1x x x approaches infinity 4 8 12 100 2.4414 2.5658 2.6130 2.7048 x 365 10,000 100,000 1,000,000 1 1 x x 1 1x x x approaches infinity 4 8 12 100 365 2.4414 2.5658 2.6130 2.7048 2.7146 x 10,000 100,000 1,000,000 1 1 x x 1 1x x x approaches infinity 4 8 12 100 365 10,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 x 100,000 1,000,000 1 1 x x 1 1x x x approaches infinity 4 8 12 100 365 10,000 100,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183 x 1,000,000 1 1 x x 1 1x x x approaches infinity 4 8 12 100 365 10,000 100,000 1,000,000 2.4414 2.5658 2.6130 2.7048 2.7146 2.7182 2.7183 2.7183 x e e 1 A P 1 x x approaches infinity 1 1 x xrt x e 1 A P 1 ex x approaches infinity 1 1 x xrt x A Pe rt Continuous Compound Interest Formula A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P? r ? t ? A? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P? r ? t ? A? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P 3850 r ? t ? A? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P 3850 r 0.075 t ? A? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P 3850 r 0.075 t unknown A? A Pe rt Mrs. Johnson received a bonus equivalent to 10% of her yearly salary and has decided to deposit it in a savings account in which interest is compounded continuously. Her salary is $38,500 per year and the account pays 7.5% interest. How long will it take for her deposit to double in value? P 3850 r 0.075 t unknown A 7700 A Pe rt 7700 3850e 0.075t P 3850 r 0.075 t unknown A 7700 7700 3850e 0.075t 7700 3850e 0.075t 3850 3850 2e 0.075t ln 2 ln e 0.075t ln 2 0.075t ln e ln 2 0.075t ln e ln e ? ln 2 0.075t ln 2 t 0.075 To the nearest year ln 2 0.075t ln e ln 2 0.075t ln 2 t 0.075 9t To the nearest year A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P? r ? t ? A? A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P 2000 A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P 2000 r unknown A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P 2000 r unknown t 20 A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P 2000 r unknown t 20 A 10,000 A Pe rt Following the birth of child, a parent wants to make an initial investment of $2000 and would like it to grow to a value of $10,000 by the Childs 20th birthday. What annual interest rate compounded continually should the parent look for? P 2000 r unknown t 20 A 10,000 A Pe rt 10,000 2000 e P 2000 r unknown t 20 r 20 A 10,000 10,000 2000 e 20 r 10,000 2000 e 2000 2000 5e 20 r ln 5 ln e 20 r 20 r ln5 ln e 20 r ln5 20r lne ln e ? ln5 20r ln5 20r ln 5 r 20 Nearest hundredth of a percent ln5 20r ln 5 r = 0.0805 20 8.05% How long will it take to double and investment at 4.5% interest How long will it take to double and investment at 4.5% interest A Pe P? r ? rt t ? A? How long will it take to double and investment at 4.5% interest A Pe P? rt How long will it take to double and investment at 4.5% interest A Pe PP r ? rt How long will it take to double and investment at 4.5% interest A Pe PP r 0.045 rt t ? How long will it take to double and investment at 4.5% interest A Pe PP r 0.045 rt t unknown A? How long will it take to double and investment at 4.5% interest A Pe PP r 0.045 rt t unknown A 2P How long will it take to double and investment at 4.5% interest A Pe rt 2P Pe 0.045t PP r 0.045 t unknown A 2P How long will it take to double and investment at 4.5% interest 2P Pe 0.045t Solve for t to the nearest hundredth 2P Pe 0.045t P P 2e 0.045t ln 2 ln e 0.045t ln 2 ln e 0.045t ln e ? ln 2 0.045t ln e ln 2 0.045t ln 2 0.045t ln 2 0.045 t How long will it take to double and investment at 4.5% interest ln 2 t 0.045 Solve for t to the nearest hundredth t 15.40 years What interest rate should an investor seek to double his money in 20 years? You solve for r to the nearest tenth of a percent A Pe rt What interest rate should an investor seek to double his money in 20 years? You solve for r to the nearest tenth of a percent ln 2 r 20 r 0.035 3.5% Will invests $2000 in a bond trust that pays 9% interest compounded semiannually. His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously. Who will have more money after 20 years, Will or Henry? How much more money? Will invests $2000 in a bond trust that pays 9% interest compounded semiannually. .09 A 2000 1 2 220 11,632.73 His friend Henry invests $2000 in a Certificate of Deposit that pays 8 ½ % compounded continuously. .08520 A 2000e 10,947.89 Who will have more money after 20 years, Will or Henry? How much more money? $684.84 At the age of 25 Coris invests $2000 in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires. Pat, who is 35, also invests $2000 in an IRA. Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65. To the nearest hundred, how much more money does Coris collect than Pat? At the age of 25 Coris invests $2000 in an Individual Retirement Account (IRA) that is allowed to accumulate interest tax-free until she retires. Pat, who is 35, also invests $2000 in an IRA. Pat and Coris each earn 8% annually compounded continuously, and each withdraws the funds from the account at age 65. To the nearest hundred, how much more money does Coris collect than Pat? Link $27,000 Back to Table of Contents P P0e rt P P0e P0 rt is the size of the original population P P0e rt P is the size of the population after t years P P0e rt r annual growth rate of the population P P0e rt t is the number of years The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region. b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit. The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 a) Using these statistics, find the average growth rate of the population in Raleigh-Durham (to the nearest tenth of a percent) and determine an equation of the population growth curve for this region. b) Assuming a constant growth rate, predict the population for Raleigh-Durham in 1995 to the nearest unit. P P0e rt Find r P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 r P t P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 r P t P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 P 560,774 r t P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 560,774 r P 665,400 t P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 560,774 r P 665,400 t unknown P P0e rt The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 P0 560,774 r unknown P 665,400 t 7 P P0e rt Find the population growth rate for Raleigh-Durham, North Carolina to the nearest tenth of a percent P0 560,774 r unknown P 665,400 t 7 P P0e rt r 7 665,400 560, 774e r unknown P0 560,774 P 665,400 t 7 P P0e rt 665,400 560, 774e 665, 400 7r e 560,774 r 7 665, 400 7r e 560,774 ln ln 665, 400 560, 774 665, 400 560, 774 ln e 7r 7r ln e ln ln 665, 400 560, 774 665, 400 560, 774 665, 400 ln 560, 774 7 7r ln e 7r r 665, 400 ln 560, 774 7 r M log b log b M log b N N ln 665,400 ln 560,774 r 7 ln 665,400 ln 560,774 r 7 ln 665,400 ln 560,774 r 7 ln 665,400 ln 560,774 r 7 r 0.024 r 2.4% Nearest tenth of a percent P P0e rt We need to find an equation that MODELS population growth for Raleigh-Durham, North Carolina r 0.024 the P 560,774e 0.024 t This is the equation that MODELS the population growth in Raleigh-Durham, North Carolina r 0.024 P 560,774e 0.024 t Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit P 560,774e 0.024 t Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit To find P we need to know the value of ? t P 560,774e 0.024 t Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit The population in Raleigh-Durham, North Carolina, grew from 560,774 in 1980 to 665,400 in 1987 t = 15 P 560,774e 0.024 t Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit 0.02415 P 560,774e P 560,774e 0.024 t Use this equation to project the population in Raleigh-Durham, North Carolina in 1995 to the nearest unit P 803,774 Link Back to Table of Contents A A0 2 t d A A0 2 A0 t d is the size of the Original Population Bacteria Culture of Yeast ETC. A A0 2 t d A is the size of the Population after t time Years Weeks Minutes Months Hours ETC. Seconds A A0 2 t d d is the amount of time it takes for the population to double Years Weeks Minutes Months Hours ETC. Seconds A A0 2 t t d is the time period Years Weeks Minutes Months Hours ETC. Seconds A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A? A0 ? d ? t ? A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A A0 ? d ? t ? A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A A0 A0 d ? t ? A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A A0 A0 d 12 hrs t ? A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A A0 A0 d 12 hrs t 2 days A A0 2 t d A certain bacteria population doubles in size every 12 hours. By how much will it grow in 2 days? A A A0 A0 d 12 hrs t 48 hrs A A0 2 A A t d A0 A0 d 12 hrs t 48 hrs A A0 2 A A0 2 A A t d 48 12 A0 A0 d 12 hrs t 48 hrs A A0 2 A A0 2 A A t d 48 12 A0 A0 d 12 hrs t 48 hrs A A0 2 48 12 A A0 2 4 A A0 16 A A0 16 A 16 A0 Link Back to Table of Contents The population is 16 times greater than the size of the original population The population grows by a factor of 16 in 2 days y= b x y= 2 x y= b x x æ1 ö y = çç ÷ ÷ ÷ çè 2 ø y= 2 x y= b x x æ1 ö y = çç ÷ ÷ ÷ çè 2 ø x y= 3 x æ1 ÷ ö y = çç ÷ çè3÷ ø y= 2 x y= b x x æ1 ö y = çç ÷ ÷ ÷ çè 2 ø x y= 3 x b> 1 æ1 ÷ ö y = çç ÷ çè3÷ ø y= 2 x y= b x x æ1 ö y = çç ÷ ÷ ÷ çè 2 ø x y= 3 x b> 1 æ1 ÷ ö y = çç ÷ çè3÷ ø 0< b< 1 y= 2 x y= b x x æ1 ö x ÷ ç yy ==ççè22ø÷÷ x x æ1 ÷ ö- x yy ==çççè33÷÷ø x - x y= 3 y= b b> 1 y= b 0< b< 1 y = ab x a is a constant y = ab x - x y = ab Determine whether each equation represents an exponential growth or an exponential decay curve. You may want check your answers with your graphing calculator x 1) y = 5(2) - 0.2 t 2) y = 5(2) 3) y = 100e 0.2t - 0.31t 4) y = 100 - 100e Determine whether each equation represents an exponential growth or an exponential decay curve. You may want check your answers with your graphing calculator x 1) y = 5(2) Link Back to Table of Contents - 0.2 t 2) y = 5(2) 3) y = 100e 0.2t - 0.31t 4) y = 100 - 100e A 1 A0 2 t h A 1 A0 2 A A0 2 t h t 1 h A A0 2 t h A A0 2 t h A0 is the original quantity of the radioactive substance (isotope) A A0 2 t h A is the amount of radioactive substance after t years A A0 2 t h h is the half-life of the radioactive substance A A0 2 t h t is the number of years A A0 2 t h The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years. A A0 2 t h The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years. A0 t A h A A0 2 t h The half life of radioactive radium (226Ra) is 1599 years. Given an initial quantity of 10 grams, how much will remain after 1000 years. A0 10 t 1000 A h 1599 Unknown A 10 2 1000 1599 A 10 2 A 6.48 1000 1599 grams Link Back to Table of Contents T f Tr To Tr e T0 rt is the Original temperature of the object T f Tr To Tr e Tr rt is the temperature of the surrounding air T f Tr To Tr e Tf rt is the final temperature of the object after t minutes T f Tr To Tr e r is the rate at which the object is cooling rt T f Tr To Tr e rt A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. To the nearest minute T f Tr To Tr e rt 160 68 212 68 e 0.21t A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. T f Tr To Tr e rt 160 68 212 68 e 0.21t A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. T f Tr To Tr e rt 160 68 212 68 e 0.21t A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. T f Tr To Tr e rt 160 68 212 68 e 0.21t A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. T f Tr To Tr e rt 160 68 212 68 e 0.21t A chef wants the soup to be served to the customer at a temperature of no less than 160º F. It has been determined that the cooling rate r for this soup is 0.21ºF per minute. When the soup is removed from the pot, it is 212ºF. If the room temperature in the restaurant is 68ºF, determine in how many minutes the soup must be served to meet the chef’s request. 160 68 212 68 e 160 68 144 e 68 68 92 144 e 144 0.21t 144 92 0.21t e 144 0.21t 0.21t 92 0.21t e 144 92 0.21t ln ln e 144 Law #2 Law #3 ln92 ln144 0.21t ln e Law #2 Law #3 ln92 ln144 0.21t ln e ln 92 ln144 t ln e 0.21 ln 92 ln144 t 0.21 To the nearest minute Law #2 Law #3 ln92 ln144 0.21t ln e ln 92 ln144 t ln e 0.21 ln 92 ln144 t 0.21 To the nearest minute 2 min T f Tr To Tr e Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF To the nearest hundredth of a minute rt T f Tr To Tr e rt Given that the original temperature of the coffee is 155ºF and the room temperature is 75ºF, determine after how many minutes the coffee will be 110ºF To the nearest hundredth of a minute 2.75 min Link Back to Table of Contents