Growth Decay Free Powerpoint Templates Page 1

Growth & Decay Free Powerpoint Templates Page 1

Things to remember… f(x) = bx ac a is the initial value b is the number of periods per unit of time c is the periodic multiplicative factor - how something changes over a period Free Powerpoint Templates Page 2

Example A bacteria population, containing 3 bacterium initially, doubles every four hours. a = initial value = 3 c = base = 2 b = period = 1 4 (once every four hours) Free Powerpoint Templates Page 3

Example 1 A mosquito population doubles every seven days. If there were 5 mosquitoes initially, after how many days will the population grow to 80? a = initial value = 5 y = 5(2)t/7 c = base = 2 80 = 5(2)t/7 b = period = 1/7 16 = (2)t/7 (once every seven days) 24 = (2)t/7 4 = t/7 28 = t Free Powerpoint Templates Page 4

• In general: Rate if it gets bigger: 1 + r if it gets smaller: 1 - r “r” is the rate as a decimal Time f(x) = acbx Final amount Initial amount

Example 2 How much money will you have if you invest 700$ at 7. 5% compounded annually for 15 years? f(x) = acbx f(x) = a(1+ r)bx f(x) = (700)(1. 075)15 = $2071. 21 Free Powerpoint Templates Page 6

Example 3 In a newly created wildlife sanctuary, it is estimated that the numbers of the population of deer will triple every 3. 7 years. Initially there are 46 deer, when will there be 11 178 deer? f(x) = acbx 11 178 = 46(3) x/3. 7 243 = (3)x/3. 7 35 = (3)x/3. 7 5= x 3. 7 x = 18. 5 years

Example 4 You buy a car for $25000. If it depreciates by 4. 3% annually, what’s the value of the car 10 years later? f(x) = 25000(1 -0. 043)x f(x) = (25000)(1 - 0. 043)10 f(x) = (25000)(0. 957)10 = $16108. 66 Free Powerpoint Templates Page 8

Example 5 An investment compounded annually at 6% grows to $1296 in 12 years. What was the initial investment? f(x) = a(1+0. 06)x 1296 = a(1. 06)12 1296 = a(2. 012) $644. 07 = a Free Powerpoint Templates Page 9

Example 6 An investment of $900 compounded annually grows to $1340 in 8 years. What rate of interest did the investment earn? f(x) = 900(1+ r)x 1340 = (900)(1+r)8 1. 48 =(1+r)8 8 √ 1. 48 = (1+r) 1. 051 = 1+r 0. 051 = r So 5. 1%

Example 7 A sheet, hung out to dry, loses moisture in the wind at a rate of about 60% per hour. How much moisture will remain after 5 hours? f(x) = (100)(1 - 0. 60)5 f(x) = 1. 024% Free Powerpoint Templates Page 11

Example 8 A certain substance decays exponentially over time and is modelled by the function where N(t) is the final amount of the element measured in grams and t is time in years. Find how much of the substance is present 4 000 years later. Free Powerpoint Templates Page 12