4.6 Growth and Decay 12/6/2017

Growth and Decay Agenda Objectives Go over #1-6 from yesterday Announcements/Opening Exponential Growth Formula Compound Interest Formula Exponential Decay Formula Students will be able to solve growth, decay, and compound interest problems when given the appropriate formulas.

Announcements Tonight’s Homework: page 23 of packet: 7-6 #1-3 Tomorrow and Friday we will… review for the Unit 4 assessment Monday: Unit 4 Assessment M.C. Tuesday: STAR Testing- meet in LRC Wednesday: Unit 4 Assessment F.R. Thursday: Midterm Review, make up missing assignments after school Friday: Midterm Review, last day to make up missing assignments after school

Exponential Growth (Page 21) The number of online blogs has rapidly increased in the last 15 years. In fact, the number of blogs increased at a monthly rate of about 13.7% over 21 months, starting with 1.1 million blogs in November 2003. The average number of blogs per month from 2003-2005 can be modeled by the equation 𝑦=1.1 1+0.137 𝑡 or 𝑦=1.1 1.137 𝑡 where y represents the total number of blogs in millions and t is the number of months since November 2003.

𝑦=1.1 1+0.137 𝑡 Exponential Growth The number of blogs increased at a monthly rate of about 13.7% over 21 months, starting with 1.1 million blogs in November 2003. The average number of blogs…where y represents the total number of blogs in millions and t is the number of months since November 2003. Label the equation. 𝑦=1.1 1+0.137 𝑡

𝑦=𝑎 1+𝑟 𝑡 Exp. Growth Formula In general, the equation for exponential growth is: 𝑦=𝑎 1+𝑟 𝑡

Example 1 The prize for a radio station contest begins with a $100 gift card. Once a day, a name is announced. The person has 15 minutes to call or the prize increases by 2.5% for the next day. Write an equation to represent the amount of the gift card in dollars after t days with no winners. How much will the gift card be worth if no one wins after 10 days?

Example 2 A college’s tuition has risen 5% each year since 2000. If the tuition in 2000 was $10,850, write an equation for the amount of the tuition t years after 2000. Predict the cost of the tuition for this college in 2020.

Compound Interest 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 Compound interest is a special kind of exponential growth. It is interest earned or paid both on the initial investment and previously earned interest.   In general, the equation for compound interest is as follows: 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡

Example 3 Maria’s parents invested $14,000 at 6% per year compounded monthly. How much money will there be in the account after 10 years?

Example 4 Determine the amount of an investment if $300 is invested at an interest rate of 3.5% compounded every other month for 22 years.

Exponential Decay 𝑦=𝑎 1−𝑟 𝑡 In general, the equation for exponential decay is as follows:   𝑦=𝑎 1−𝑟 𝑡

Example 5 A fully inflated child’s raft for a pool is losing 6.6% of its air every day. The raft originally contained 4500 cubic inches of air. Write an equation to represent the loss of air. Estimate the amount of air in the raft after 7 days.

Example 6 The population of Campbell County, Kentucky has been decreasing at an average rate of about 0.3% per year. In 2000, its population as 88,647. Write an equation to represent the population since 2000. If the trend continues, predict the population in 2018.

EXTRA EXAMPLES

Is it growth, decay, or compound interest? Mason invested $600 into an account with a 2.3% interest rate compounded monthly. How much will his investment be worth in 5 years?

Is it growth, decay, or compound interest? Ms. Caudill purchases a new car for $18,750. The car depreciates at a rate of 20% annually. After 7 years, how much will the car be worth?

Is it growth, decay, or compound interest? A certain radioactive isotope decays at a rate of 5% every year. If we started with 10 grams of this radioactive material, how much will be left in 100 years?

Is it growth, decay, or compound interest? A certain bacteria population doubles in size every 5 minutes. If we start with a culture of 5 bacteria, how many bacteria will there be in an hour?

Is it growth, decay, or compound interest? A city’s population is 1,678,972 and is decreasing at an annual rate of 0.1%. What is the population after 15 years?

Is it growth, decay, or compound interest? In the year 2007, a scientist determined that there were 300 of a certain type of frog in a pond. The population increased exponentially at a rate of 8% each year. Based on this information, what is the frog population going to be in the year 2025?