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Vs. Vs. The M i l l i o n D o l l a r Mission

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The Situation You’re sitting at your desk in the middle of math class and in walks Bill Gates. He’s looking to hire you for his next mission. He’s going to need you to work for the next 30 days. (Oh no! You’re going to miss a whole month of school! I know…you’re just crushed!) He didn’t say exactly what the mission was, but he assured us that it was not too dangerous, and the majority of it would take place on the island of Tahiti.

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The Decision He offers you a choice of two payment options. You can choose either: One cent on the first day, two cents on the next day, and he will double your salary everyday for the rest of your mission (30 days) One cent on the first day, two cents on the next day, and he will double your salary everyday for the rest of your mission (30 days)OR One Million Dollars (that’s $1,000,000!!!!!) One Million Dollars (that’s $1,000,000!!!!!) What is your first reaction?

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Your Classroom Mission You need to prove to me that you made the right decision! You need to do this in some sort of organized way Use of a chart, graph, formula, etc. You must write down every step that you took to come up with your answer. Work in pairs to come up with how much money you would make if you chose option one. Compare your answer with the second option to see if you made the right decision.

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Now let’s find out for sure if you made the right decision! Here is your salary with option #1: Pay with First Option - Week 1 Day No.Pay for that DayTotal Pay (In Dollars) 1.01 2.02.03 3.04.07 4.08.15 5.16.31 6.32.63 7.641.27 * The charts and some excerpts taken from http://math.rice.edu/%7Elanius/pro/rich.htmlhttp://math.rice.edu/%7Elanius/pro/rich.html

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So you've worked a whole week and only made $1.27. That's pretty awful, all right. There's no way to make a million in a month at this rate…right? Let's check out the second week: Pay with First Option - Week 2 Day No.Pay for that DayTotal Pay (In Dollars) 81.282.55 92.565.11 105.1210.23 1110.2420.47 1220.4840.95 1340.9681.91 1481.92163.83

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Well, he would make a little more the second week, at least he's over $100. But there's still a big difference between $163.83 and $1,000,000. Want to see the third week? Pay with First Option - Week 3 Day No.Pay for that DayTotal Pay (In Dollars) 15163.84327.67 16327.68655.35 17655.361,310.71 181,310.722,621.43 192,621.445,242.87 205,242.8810,485.75 2110,485.7620,971.51

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We're getting into some serious money here now, over $20,000, but still nowhere even close to a million. And there's only 10 days left. So it looks like the million dollars is the best deal. Of course, we suspected that all along. Pay with First Option - Week 4 Day No.Pay for that DayTotal Pay (In Dollars) 2220,971.5141,943.03 2341,943.0483,886.07 2483,886.08167,772.15 25167,772.16335,544.31 26335,544.32671,088.63 27671,088.641,342,177.27 281,342,177.282,684,354.55

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Hold it! Look what has happened. What's going on here? We went from $21,000 to over a million in 6 days. This can't be right. This is amazing. Look how fast this pay is growing. Let's keep going. I can't wait to see what the total will be! Pay with First Option Day No.Pay for that DayTotal Pay (In Dollars) 292,684,354.565,368,709.11 305,368,709.1210,737,418.23 In 30 days, it increases from 1 penny to over 10 million dollars. That is absolutely amazing!

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This problem is an example of exponential growth. Your salary is growing exponentially. Have you ever heard of anything growing exponentially before?

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Can you think of another way to figure this out? Perhaps we can do this without making a table from day 1 to day 30? Let’s see… On day 1 the number stayed the same, we earned one cent. On day 1 the number stayed the same, we earned one cent. –So let’s say on day one we multiplied $0.01 by 1. On day 2 we multiplied day 1’s pay times 2. On day 2 we multiplied day 1’s pay times 2. –That’s $0.01(1 x 2) On day 3 we multiplied day 2’s pay times 2. On day 3 we multiplied day 2’s pay times 2. –That’s $0.01(1 x 2 x 2) On day 4 we multiplied day 3’s pay times 2. On day 4 we multiplied day 3’s pay times 2. –That’s $0.01(1 x 2 x 2 x 2)

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So we are multiplying by another 2 for each day. How do we write that? Let’s work backwards… For day 4, instead of writing For day 4, instead of writing $0.01(1 x 2 x 2 x 2) we can write $0.01 x 2 3 For day 3, instead of writing For day 3, instead of writing $0.01(1 x 2 x 2), we can write $0.01 x 2 2 For day 2, instead of writing For day 2, instead of writing $0.01(1 x 2), we can write $0.01 x 2 1 For day 1, instead of writing For day 1, instead of writing $0.01(1), we can write $0.01 x 2 0 Can you come up with a formula for that?

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Let’s use a ‘d’ to represent the day of the paycheck: $0.01 x 2 d-1 Let’s check to make sure this works… Remember, on the 30th day, according to the chart we should have a paycheck worth 5,368,709.12. So let’s try $0.01 x 2 30-1 = $0.01 x 229 = 5,368,709.12. Great! We found out how much our paycheck would be on the 30th day. Now can you come up with a formula to represent the total amount of money you earned over the 30 days?

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Let’s look at the amount of our payment for the 30th day. Let’s look at our total salary on the 29th day. They are almost the same. They only differ by one penny! So, how can we change our formula to match the total pay? We have: $0.01 x 2 d-1 So the total pay on day 29 is going to be $0.01 x 2 29 - $0.01 Lets check to be sure $0.01 x 2 29 - $0.01 = 5,368,709.11 Great! Now let’s use ‘d’. We have $0.01 x 2 d - $0.01 Just to be sure, let’s pick a day and see if this really does work. On the 30th day we have: $0.01 x 2 30 - $0.01 = $10,737,418.23. Perfect! Day No.Pay for that DayTotal Pay (In Dollars) 292,684,354.565,368,709.11 305,368,709.1210,737,418.23

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Let’s use the formula that we found to graph the equation. In your graphing calculators let’s type in:.01 x 2d -.01 You should get something that looks like this:

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If we change the window size we can adjust it to fit the numbers we used in our problem and we’ll get:

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In 30 days, it increased from 1 penny to over 10 million dollars. Notice how the function grows very slowly at first, then increases rapidly. In 30 days, it increased from 1 penny to over 10 million dollars. Notice how the function grows very slowly at first, then increases rapidly. –Remember how it seemed to take us forever to get to $20,000, but then how quickly it grew to over a million. This is what happens with exponential growth. The rate of change increases over time, it start out r e a l l y s l o w and then all of a sudden it speeds up and gets faster and faster as x increases (or as time goes on). This is what happens with exponential growth. The rate of change increases over time, it start out r e a l l y s l o w and then all of a sudden it speeds up and gets faster and faster as x increases (or as time goes on).

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Exponential functions are of the form: Note: The variable x is an exponent. This is what makes it an exponential function! Exponential growth occurs when b > 1. In our example b = 2, so we had exponential growth. If b < 1 we have what we call exponential decay. With exponential decay the rate of change decreases over time, the rate of decay becomes slower and slower as time goes on. What would happen if b = 1? Try it in your calculators: y = 2*1 x You get the line y = 2, a straight line. This is not exponential growth or decay.

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Example: when a > 0 and the b is between 0 and 1, the graph will be decreasing (decaying). For this example, each time x is increased by 1, y decreases to one half of its previous value. Such a situation is called Exponential Decay. Such a situation is called Exponential Growth. when a > 0 and the b is greater than 1, the graph will be increasing (growing). For this example, each time x is increased by 1, y increases by a factor of 2. Example:

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Other examples of exponential functions: Label each as growth or decay: Label each as growth or decay: –Folding a piece of paper in half over and over again Did you know that if you could fold an ordinary piece of paper in half 17 times it would be taller than your average house? (http://raju.varghese.org/articles/powers2.html) http://raju.varghese.org/articles/powers2.html –Population increase Visit http://www.otherwise.com/population/exponent.html for a demonstration of this http://www.otherwise.com/population/exponent.html –Radioactive decay –Population decrease Aids Plaques Other incurable diseases Can you come up with any examples on your own? Can you come up with any examples on your own?

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When a quantity grows by a fixed percent at regular intervals, the pattern can be represented by the functions: a = initial amount before measuring growth/decay r = growth/decay rate (often a percent) x = number of time intervals that have passed GrowthDecay *You may want to keep these formulas in mind as you complete your homework problems.

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Try it! #1 In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994? (Don't consider a fractional part of a person.) * ‘Try it!’ problems taken from regentsprep.org

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Try it! #2 Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds? * ‘Try it!’ problems taken from regentsprep.org

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