Add three consecutive letters of the alphabet to the group of letters below, without splitting the consecutive letters, to form another word. DY

STUDY

Which month comes next? January, March, June, October, March, ?

September

All widgets are green. Everything green has a hole in the middle All widgets are green. Everything green has a hole in the middle. Some things that are green have a jagged edge. Therefore: All widgets have a hole in the middle Everything with a jagged edge is a widget. Neither of the above is true. Both the above are true.

1. All widgets have a hole in the middle

Exponential Modeling Week 2

Review: linear equations Your sister is selling Girl Scout cookies for $2.60 a box. She’s already made $15.60. Write an equation for the situation above: y = 2.60x + 15.60 How much does she make after selling 10 more boxes? y = 2.60 (10) + 15.60 = $41.60

Review: linear equations How many boxes does she have to sell to make $130? (y = 2.60x + 15.60) 130 = 2.60x + 15.60 (subtract 15.60 from both sides) 114.40 = 2.6x (divide both sides by 2.6) x = 44

Review: linear equations y=5x-9 Solve for y when x = 2. y = 5(2)-9 y = 1 Solve for x when y = 6. 6 = 5x – 9 (next: add 9 to both sides) 15 = 5x (next: divide both sides by 5) x = 3

Last week… We learned about linear modeling. Model: y = mx +b Graph: straight line

This week… Exponential Modeling x is an exponent.

This week… Exponential growth Exponential decay Model: Model: y = a(1+r)x y: final a: initial r: rate x: time Model: y = a(1-r)x y: final a: initial r: rate x: time

Graph:

Examples of Exponential Growth Populations tend to growth exponentially not linearly When an object cools (e.g., a pot of soup), the temperature decreases exponentially toward the ambient temperature (the surrounding temperature) Radioactive substances decay exponentially Bacteria populations grow exponentially Money in a savings account with at a fixed rate of interest increases exponentially Viruses and even rumors tend to spread exponentially through a population (at first) Anything that doubles, triples, halves over a certain amount of time Anything that increases or decreases by a percent

Difference: How can you tell? Linear: Constant Rate of Change Exponential: Constant Percent Change How can you tell? Linear, if it increases by the same or decreases by the same Exponential, calculate the percent change and see if it stays constant Percent change = (changed- reference)/reference

Percent Change example exponential growth?

Percent Change example

Percent Change example

Percent Change example The percent change is 20% each time. So it is an exponential function.

Exponential Modeling Example Two bosses A: one million dollars for one month B: a penny doubled every day for a month Who would you work for?

Exponential Modeling Example Boss B Day 1: $.02 Day 2: $.04 Day 3: $.08 … Day 10: $10.24 Day 20: $10,485.76 Change of mind?

Exponential Modeling Example Calculations y = a (1+r)x y = .01 (1+1)30 y = 10,737,418.24

Exponential Modeling Growth (savings account) y = a(1+r)x Decay (radioisotope dating) y = a(1-r) x

Exponential Growth -- Excel If roaches grow at a rate of 25% every 10 days, how long will it take 400 roaches to become 1000 in number?

Exponential Growth -- Excel

Exponential Growth -- Excel

Exponential Growth -- Excel Drag it down Right click on the + at the right bottom Move down with the mouse

Exponential Growth -- Excel

Exponential Growth -- Excel A little bit after 40 days

Exponential Decay -- Excel Dead Sea Scrolls have about 78% of the normally occurring amount of Carbon 14 in them.  Carbon 14 decays at a rate of about 1.202% per 100 years. How old are the Dead Sea Scrolls?

Exponential Decay -- Excel Since we know the rate of decay per every 100 years, make the excel table have intervals of 100 years. (after entering 0 and 100, you can select the two and drag down)

Exponential Decay -- Excel

Exponential Decay -- Excel

Exponential Decay -- Excel Answer: between 2000 to 2100 years old

Assignments: Activity 2, 3 Homework 2