Growth & Decay

1. Growth & Decay

2. f(x) = acbx Things to remember… 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

3. 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)

4. 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 c = base = 2 b = period = 1/7(once every seven days) y = 5(2)t/7 80 = 5(2)t/7 16 = (2)t/7 24 = (2)t/7 4 = t/7 28 = t

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

6. Example 2 How much money will you have if you invest 700$ at 7.5% compounded annually for 15 years? f(x) = (700)(1.075)15 = $2071.21 f(x) = acbx f(x) = a(1+ r)bx

7. 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? 243 = (3)x/3.7 35 = (3)x/3.7 5 = x 3.7 f(x) = acbx 11 178 = 46(3) x/3.7 x = 18.5 years

8. 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)10 f(x) = (25000)(0.957)10 = $16108.66 f(x) = 25000(1-0.043)x

9. Example 5 An investment compounded annually at 6% grows to $1296 in 12 years. What was the initial investment? 1296 = a(1.06)12 1296 = a(2.012) $644.07 = a f(x) = a(1+0.06)x

10. Example 6 An investment of $900 compounded annually grows to $1340 in 8 years. What rate of interest did the investment earn? 1340 = (900)(1+r)8 1.48 =(1+r)8 8√1.48 = (1+r) f(x) = 900(1+ r)x 1.051 = 1+r 0.051 = r So 5.1%

11. 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%

12. 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.