Chapter 10.6

Formulas to Memorize:

Exponential G & D

x

x

ray

aby

)1(

Continuous Exponential G & D

ktaey

y = final amounta = original amountk = rate or constantt = time

Example 1 Exponential Decay

a) A cup of coffee contains 140 mg of caffeine. If caffeine leaves the body at 10% per hour, how long will it take for half of the caffeine to be eliminated from ones body?

xray )1(

x)10.1(14070

6.5 hours

b) A particular car is said to depreciate 15% each year. Ifthe car new was valued at $20,000, what will it be worthafter 10 years?

10)15.1(20000 y$3,937.4910)85(.20000y

Example 2Example 2 Exponential Growth

b) A bacteria colony started with 2 bacteria. The colony is growing exponentially at 10% per hour. How long will it take to have 1 million bacteria?

a) A town originated in 1910 with 200 people. Since its origination,it has grown exponentially 15% per year. If the population continues to grow at this rate, what will the population be in2010?

xray )1(

100)15.1(200 y 234,862,690 people

x)10.1(2000,000,1 x)10.1(000,500

000,500log 10.1x

137.7 hours

Example 3 Continuous Exponential Growth

b) Jill invested $1000 into a CD account that claims to compound continuously at 10%. How long will it take triple her investment?

a) A population of rabbits is growing continuously at 20% eachmonth. If the colony began with 2 rabbits, how many monthsuntil they reach 10,000 rabbits?

ktaey

te 20.2000,10 te 20.5000 t20.)5000ln(

20./)5000ln(t

42.6 months

te 10.10003000 te 10.3 t10.3ln

10./)3ln(t

11.0 years

Example 4 Continuous Exponential Decay

a) The formula for the half life of Carbon-14 is given below.A paleontologist finds the bones of a Wooly Mammoth. Sheestimates the bones only contain 4% of the Carbon-14 thatit would have had alive. Estimate the age of the Mammoth.

ktaey

taey 00012.

b) The half life of Sodium-22 is given below. A geologist is studying a meteorite and estimates that it contains only 12% of as much Sodium-22 as it would have when it reached the earths atmosphere. How long ago did the meteorite reach earth.

taey 2666.

taea 00012.04. te 00012.04. t00012.)04ln(.

26,824.0 years old

taea 2666.12. t2666.)12ln(. 8.0 years old