Bellringers 4 16 20 If a group starts

Bellringers 4 -16 -20 If a group starts with 2, 000 people and it grows 7. 5% per year, how many will there be in 7 years? y = a(1 + r)ᵀ A = 2000(1 +. 075)⁷ A = 3, 318 people ANSWER Jim’s investment of $7, 500 has been losing its value at a rate of 2. 5% each year. What will his investment be worth in 6 years if it continues to decrease like this? Y = a(1 – r)ᵀ Y = 7500(1 -. 025)6 Y = 7500(. 975)6 Y = $6, 443. 01 ANSWER

Section 7 -6 More Growth & Decay Previously, we said y = ab�was the formula for exponential growth. Another formula for exponential growth is: y = a(1 + r)ᵀ, where “a” is the initial amount, r is the rate of change, and T is the time, usually in years. r = % changed to decimal 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 and win. If they don’t win, then 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. y = a(1 + r)ᵀ becomes y = 100(1 +. 025)ᵀ where initial = 100 and 2. 5% =. 025 for r

y = 100(1 +. 025)ᵀ How much will the prize be if no one wins after 10 days? y = 100(1 +. 025)¹⁰ y = 100(1. 025)¹⁰ y = $128. 01 ANSWER 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 tuition at this school in 2015. y = 10, 850(1 +. 05)ᵀ starting tuition: $10, 850 increase of 5% which is. 05 as a decimal, over T years y = 10, 850(1 +. 05) 15 y = $22, 556. 37 ANSWER

Equation for Compound Interest (exponential growth) Equation for compound interest: A = P(1 + r/n)^(nt) Where P is the principal or initial amount, r is the interest rate as a decimal, n is the number of times interest is compounded each year, and t is time in years. # word 1 annually 12 monthly 2 semi-annually 52 weekly 4 quarterly 365 or 360 daily

A = P(1 + r/n)^(nt) Bob’s parents give him $10, 000 invested at 7% compounded monthly. How much money will he have in 10 years? A = 10, 000(1 +. 07/12)^(12*10) A = $20, 096. 61 ANSWER after 10 years What if he left it there for 30 years under those conditions? A = 10, 000(1 +. 07/12)^(12*30) A = $81, 164. 97 ANSWER after 30 years Marla’s parents invested $23, 000 at 6% per year compounded weekly. Write equation for amount after T years. If her parents gave this to her when she was 23, find the amount it will be worth when she is 59. A = 23, 000(1 +. 06/52)^(52*36) A = $199, 187. 98 ANSWER after 36 years

Exponential Decay In exponential decay, the original amount decreases by the same percent over a period of time. A variation of the growth equation can be used as the general equation for exponential decay. Y = a(1 – r)ᵀ where a is the initial amount, r is the rate of change expressed as a decimal, and T is time (usually number of years). A fully inflated child’s raft is losing 6. 6% of its air every day. The raft originally contained 4500 cubic inches of air. This time T is time in days. Write an equation to represent the loss of air. y = 4, 500(1 -. 066)ᵀ y = 4, 500(. 934)ᵀ y is cubic inches of air left T days.

Previous formulas y = 4, 500(1 -. 066)ᵀ y = 4, 500(. 934)ᵀ Using this formula, estimate the amount of air in the raft after 7 days. y = 4, 500(. 934)⁷ y = 2, 790. 23 cubic inches ANSWER The population of Campbell County, Kentucky has been decreasing at an average rate of 0. 3% per year. In 2000, its population was 88, 647. Write an equation to represent the population since 2000. If the trend continues, predict the population in 2010. y = 88, 647(1 -. 003)10 y = 86, 023 people ANSWER

Ms Acosta received a job as a teacher with a starting salary of $34, 000. According to her contract, she will receive a 1. 5% increase in her salary every year. How much will she make in 7 years? y = a(1 + r)ᵀ y = 34000(1 +. 015)⁷ y = 34000(1. 015)⁷ y = $37, 734. 73 ANSWER Paul invested $400 in an account with a 5. 5% interest compounded monthly. How much will his investment be worth in 8 years? A = P(1 + r/n)^(nt) A = 400(1 +. 055/12)^(12*8) A = 400(1. 004583333)^96 A = $620. 46 ANSWER

In 2000, 2200 students attended Pollywog High School. The enrollment has been declining 2% annually. Write an equation for the enrollment there T years after 2000. If this trend continues, how many students will be enrolled in 2015? Y = a(1 – r)ᵀ Y = 2200(1 –. 02)ᵀ Y = 2200(. 98)15 Y = 1, 625 students in 2015 ANSWER