Accelerated Math II. Exponential Applications Review. One-Minute Question. The measure of A is 30 and the sin D = 3/5. If BA = 18, find DC. C D B A.
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Accelerated Math II Exponential Applications Review
One-Minute Question The measure of A is 30 and the sin D = 3/5. If BA = 18, find DC. C D B A
One-Minute Question The measure of A is 30 and the sin D = 3/5. If BA = 18, find DC. C CB = 18/sqrt(3) = 6sqrt(3) 3/6 = 6sqrt(3)/x, so X = 10sqrt(3) D B A
Personal Responsibility for Learning • A) Do your homework • B) Check your homework – either with the answers provided (by the book, the blog or other teacher’s blogs) or with other students. • C) Check for your own mistakes. • D) Listen when problems are being worked. • E) Don’t ask to see a problem that was already worked. • F) Be courteous to others.
Sample problems • Write the equation for the exponential function through (3, 6) and (1, 2). • Y = 1.1547(1.732)x • Does this model growth or decay? • Growth • What is the growth rate? • 73.21%
Sample problems • Sam invested $25000 at 4.2% interest • If it was compounded yearly for 5 years, how much money will he make? • If it was compounded monthly for 5 years, how much money will he make? • If it was compounded continually for 5 years, how much money will he make?
Sample problems • Sam invested $2500 at 4.2% interest • If it was compounded yearly for 5 years, how much money will he make? $3070.99 • If it was compounded monthly for 5 years, how much money will he make? $3083.06 • If it was compounded continually for 5 years, how much money will he make? $3084.20
Sample problems • A radioactive substance has a half life of 49 days. How much of this substance will still be radioactive after 1 year? • (1/2)365/49 = .005723 or .5723%
Predicting School Populations: In 2000, the high school population in Pecan County was 1500 students. 2012 Summer Academy In 2010, the high school population had increased to 2000 students.
Predicting School Populations: By what percent did the population increase over this 10 year period? 2012 Summer Academy What is the growth factor for the population over this decade?
Answer: • (2000-1500)/1500 = 33 1/3 % • There are other ways to get this answer as well, but do realize that it asks for the rate over the 10 year period and not for the yearly rate of increase.
School Populations: Pecan County builds high schools to house 1600 students, but by using trailers, that capacity can be increased by 50%. Then a new high school must be built and opened. 2012 Summer Academy
School Populations: If the School Board assumes that population growth is linear, find an equation to model Pecan County’s high school population. 2012 Summer Academy In what year should the School Board plan on opening a new high school?
Answer: • L(t) = 50t + 1500 , where t is the number of years after 2000. • Solve 50t + 1500 = 2400 (why 2400?) • 50t = 900, so t = 900/50 = 18 • New school should open in 2018.
School Populations: Suppose that the population growth is actually exponential. Find a new equation to model Pecan County’s high school population. 2012 Summer Academy Based on this model, when should a new high school be opened?
Answer • E(t) = 1500*1.029186^t • Solve 1500*1.029186^t = 2400 • I’d graph each side or else look at the table. • New school should open in 2016. (Actually the population reaches 2400 when t= 16.338, so both 2016 and 2017 can be argued since we aren’t told at what time during the year the population is taken.)
Suppose that the population growth is actually exponential, but the school board used the linear model to determine when the new school should be built. School Populations: 2012 Summer Academy Use mathematics to describe the population just before the new school opens.
Answer • When the new school opens in 2018 based on the linear model, the actual school population will be 2517 growing exponentially. The school will be 117 over maximum capacity, and will have been over capacity for almost two years.
Remember the Rules of Exponents • xmxn = xm+n • (xm)n = xm n • x0 = 1 • x-n =