8.3 Applications of Exponential Functions

3/12/2014

Growth by doubling: Bacteria

One of the most common examples of exponential growth deals with bacteria. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. For example, if we start with only one bacteria which can double every hour, by the end of one day how many bacteria will we have?

http://www.regentsprep.org/regents/math/algebra/ae7/expdecayl.htm

End of  Hour 1 2 3 4 5 6 7 8 9 10 ... 24

Bacteria - starting with

one2 4 8 16 32 64 128 256 512 1024 ... 16,777,216

Pattern: 21 22 23 24 25 26 27 28 29 210 224

Compound InterestInterest that builds up on the initial principal and the accumulated interest of a principal deposit, loan or debt.

Compounding Interest Formula

Where P(t) = amount of money accumulated after t years, including interest. Po = principal amount (the initial amount you borrow or deposit) r = annual rate of interest (as a decimal)n = number of times the interest is compounded per year t = number of years the amount is deposited or borrowed for.

An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4%, compounded quarterly. What is the balance after 6 years?

Po = $1,500

r = .04n = 4 (quarterly = 4 times per year)t = 6 yrs

= = $1,904.60

Calculator: Follow order of Operations: Do what’s in the ( ), then raise it to the exponent, then multiply by 1500.

Exponential Growth Formula

Where P(t) = the amount of substance after time tPo = initial/starting amount

b = growth factor = 2 for doubling = 3 for triplingt = time elapsedr = time it takes for growth to occur.

Sarah observes that the number of bacteria in the colony in the lab doubles every 30mins. If the initial number of bacteria in the colony is 50, what is the total number of bacteria in the colony after 5 hours?

Po = 50

b = 2 t = 5 hrsr = .5 hrs (30mins)

𝑃 (𝑡 )=𝑃0(𝑏)𝑡𝑟

= =51,200 bacteria

After 5 hrs, there are 51,200 bacteria

Calculator: raise 2 to (5÷0.5) then multiply by 50.

Half Lifeis the amount of time that the substance's total amount is halved.

Exponential Decay Formula (half- life)

Where P(t) = the amount of substance left after time tPo = initial/starting amount

d = decay factor = ½ for half-lifet = time elapsedr = time it takes for decay to occur.

Technitium-99m is a radioactive substance used to diagnose brain, thyroid liver and kidney diseases. This radioactive substance has a half life of 6 hours. If there are 200 mgs of this technetium-99m, how much will there be in 12 hours?

Po = 200 mg

d = ½ t = 12 hrsr = 6 hrs

𝑃 (𝑡 )=𝑃0(𝑑)𝑡𝑟

= = 50mg

After 12 hrs, there’s 50mg left.

Calculator: raise 0.5 to (12÷6 or 2) then multiply by 200.

Homework:

WS 8.3 do ALL