Doubling Time and Half-Life Chapter 8 Unit B Doubling Time and Half-Life

Exponential Growth/Decay Formula 𝑛𝑒𝑤 𝑣𝑎𝑙𝑢𝑒 =𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑥 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑜 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦

Exponential Growth/Decay Formula Ex 1: new value = 500 x 3^(t/8) Imagine that the unit t is years. Then, this equation models a situation where a quantity begins at ______ and ______ every ______ years.

Exponential Growth/Decay Formula Example 2: new value = 40 x (0.5)^(t/4) Imagine that t is in minutes. Then this equation models a situation where a quantity begins at ____ and ____ every _____ minutes.

Exponential Growth/Decay Formula Ex 3: Your bank gives you 3% annual interest on your savings account. If you begin with $1000 in your account, how much money will you have after t years? New amount = 1000 x 1.03^t Question: Why doesn’t t have a denominator?

Rule of 70 Time to double = ≈ 70 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑎𝑠 𝑎 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 Time to halve ≈ 70 𝑑𝑒𝑐𝑎𝑦 𝑟𝑎𝑡𝑒 𝑎𝑠 𝑎 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 **Works for growth/decay rates that are less than about 15%**

Rule of 70 Ex 1: If your bank interest rate is 4% how long will it take for your money to double? Ex 2: If your car is losing value at 8%, when will your car be worth half of what you bought it for?

Example: Wild Rabbits A population of wild rabbits is growing according to the formula: 𝑅=80 𝑥 2 𝑡 10 where t is in months. How long does it take for the population to double? What will the population be after 30 months? What will the population be after 10 years?

Example: Radioactive Material A radioactive material has a half-life of 250 years. Suppose we start with a 100-gram sample of this material. What is an equation that expresses the amount left after t years? How much material will be left after 50 years?

Ex: Patient A patient takes 10 mg of a drug, and 4 hours later 5 mg of the drug remains in his system. Write a formula for the amount of drug remaining in his system after t hours.

Ex: Patient #2 A patient takes 2 mg of a drug at 9 AM. Her body will clear 70% of the remaining drug every hour, so that 30% of the drug remains in her body. How much of the drug would be left in her system at noon?