Section 5.6. Applications and Models: Growth and Decay; Compound
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1 Section 5.6 Applications and Models: Growth and Decay; Compound Interest
2 Exponential Growth A quantity that experiences exponential growth will increase according to the equation P(t) = P 0 e kt t is the time (in any given units)
3 Exponential Growth A quantity that experiences exponential growth will increase according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t
4 Exponential Growth A quantity that experiences exponential growth will increase according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t P 0 is the initial quantity.
5 Exponential Growth A quantity that experiences exponential growth will increase according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t P 0 is the initial quantity. k (which needs to be positive) is the exponential growth rate.
6 Exponential Growth (continued) A quantity that experiences exponential growth also has a corresponding doubling time. If the doubling time is T, then the population will increase according to the equation P(t) = P 0 e kt, k = ln 2 T Notice you can also solve for T to get the equation T = ln 2 k
7 Examples 1. The exponential growth rate of a population of rabbits is 11.6% per month. What is the doubling time?
8 Examples 1. The exponential growth rate of a population of rabbits is 11.6% per month. What is the doubling time? About 6 months.
9 Examples 1. The exponential growth rate of a population of rabbits is 11.6% per month. What is the doubling time? About 6 months. 2. A sample of bacteria is growing in a Petri dish. There were originally 2 thousand cells, and after 2 hours there are now 5 thousand cells. How long will it take for there to be 8 thousand cells?
10 Examples 1. The exponential growth rate of a population of rabbits is 11.6% per month. What is the doubling time? About 6 months. 2. A sample of bacteria is growing in a Petri dish. There were originally 2 thousand cells, and after 2 hours there are now 5 thousand cells. How long will it take for there to be 8 thousand cells? About 3 hours.
11 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n
12 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n P(t) is the total amount of money at time t
13 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment.
14 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment. r is the interest rate.
15 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment. r is the interest rate. n is the number of times the interest is compounded per year.
16 Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) ( ) P(t) = P r nt n P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment. r is the interest rate. n is the number of times the interest is compounded per year. Page 327 has a chart with key words to help figure out what n is.
17 Continuously Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) P(t) = P 0 e kt
18 Continuously Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) P(t) = P 0 e kt P(t) is the total amount of money at time t
19 Continuously Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) P(t) = P 0 e kt P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment.
20 Continuously Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) P(t) = P 0 e kt P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment. k is the nominal interest rate.
21 Continuously Compounded Interest An an investment earning continuously compounded interest grows according to the formula: t is the time (in years) P(t) = P 0 e kt P(t) is the total amount of money at time t P 0 is the principal - or initial amount of the investment. k is the nominal interest rate. Notice that this is exactly the same as the formula for exponential growth. Problems involving exponential growth and continuously compounded interest work exactly the same.
22 Example Suppose that $82, 000 is invested at % interest, compounded quarterly. Find the function for the amount to which the investmnt grows after t years.
23 Example Suppose that $82, 000 is invested at % interest, compounded quarterly. Find the function for the amount to which the investmnt grows after t years. P(t) = 82000( ) 4t
24 Example Suppose that $82, 000 is invested at % interest, compounded quarterly. Find the function for the amount to which the investmnt grows after t years. P(t) = 82000( ) 4t A father wishes to invest money to help pay for his son s college education. The investment earns 5% compounded continuously. How much should he invest when his son is born so that he ll have $50,000 when his son turns 18?
25 Example Suppose that $82, 000 is invested at % interest, compounded quarterly. Find the function for the amount to which the investmnt grows after t years. P(t) = 82000( ) 4t A father wishes to invest money to help pay for his son s college education. The investment earns 5% compounded continuously. How much should he invest when his son is born so that he ll have $50,000 when his son turns 18? $
26 Exponential Decay A quantity that experiences exponential decay will decrease according to the equation P(t) = P 0 e kt t is the time (in any given units)
27 Exponential Decay A quantity that experiences exponential decay will decrease according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t
28 Exponential Decay A quantity that experiences exponential decay will decrease according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t P 0 is the initial quantity.
29 Exponential Decay A quantity that experiences exponential decay will decrease according to the equation P(t) = P 0 e kt t is the time (in any given units) P(t) is the amount at time t P 0 is the initial quantity. k (which needs to be positive) is the decay rate.
30 Exponential Decay (continued) A quantity that experiences exponential decay also has a corresponding half-life. If the half-life if T, then the sample will decrease according to the equation P(t) = P 0 e kt, k = ln 2 T Notice you can also solve for T to get the equation T = ln 2 k
31 Example The half-life of radium-226 is 1600 years. Find the decay rate.
32 Example The half-life of radium-226 is 1600 years. Find the decay rate. k = ln = % per year
33 Newton s Law of Cooling An object that s hotter/colder than it s surrounding environment will cool off/heat up according to the equation T (t) = T 0 + (T 1 T 0 )e kt t is the time (in any given units)
34 Newton s Law of Cooling An object that s hotter/colder than it s surrounding environment will cool off/heat up according to the equation T (t) = T 0 + (T 1 T 0 )e kt t is the time (in any given units) T (t) is temperature of the object at time t
35 Newton s Law of Cooling An object that s hotter/colder than it s surrounding environment will cool off/heat up according to the equation T (t) = T 0 + (T 1 T 0 )e kt t is the time (in any given units) T (t) is temperature of the object at time t T 0 is temperature of the surrounding environment
36 Newton s Law of Cooling An object that s hotter/colder than it s surrounding environment will cool off/heat up according to the equation T (t) = T 0 + (T 1 T 0 )e kt t is the time (in any given units) T (t) is temperature of the object at time t T 0 is temperature of the surrounding environment T 1 is the initial temperature of the object
37 Newton s Law of Cooling An object that s hotter/colder than it s surrounding environment will cool off/heat up according to the equation T (t) = T 0 + (T 1 T 0 )e kt t is the time (in any given units) T (t) is temperature of the object at time t T 0 is temperature of the surrounding environment T 1 is the initial temperature of the object k is a constant that depends on the physical properties of the object and its surrounding. It will change based on how easily they transfer heat to each other.
38 Example A roasted turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room the temperature is 75 F. If the temperature is 150 in half an hour, what is the temperature after 45 minutes?
39 Example A roasted turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room the temperature is 75 F. If the temperature is 150 in half an hour, what is the temperature after 45 minutes? 137 F
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