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LSP 120

LSP 120. Exponential Modeling. What Makes It Exponential?. Linear relationship – where a fixed change in x increases or decreases y by a fixed amount Exponential relationship – for a fixed change in x , there is a fixed percent change in y. Exponential, Linear, or Neither?.

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LSP 120

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  1. LSP 120 Exponential Modeling

  2. What Makes It Exponential? • Linear relationship – where a fixed change in x increases or decreases y by a fixed amount • Exponential relationship – for a fixed change in x, there is a fixed percent change in y

  3. Exponential, Linear, or Neither? x y Percent change 0 192 no formula here 1 96 =(B3-B2)/B2 2 48 3 24 Apply =(B3-B2)/B2. If the column is constant, then the relationship is exponential. x y Percent change 0 192 1 96 -50% 2 48 -50% 3 24 -50% Note: Answer was -.5 but we converted these cells to % by clicking on the % icon on the toolbar

  4. Linear, Exponential or Neither? Is each of these linear, exponential, or neither? x y 5 0.5 10 1.5 15 4.5 20 13.5 x y 0 0 1 1 2 4 3 9 x y 0 0 1 5 2 9 3 13 x y 0 192 1 96 2 48 3 24 Recall: to determine if it is linear: =(B3-B2)/(A3-A2) To determine if it is exponential: =(B3-B2)/B2

  5. General Equation • As with linear, there is a general equation for an exponential function y = A * (1 + p)x where A is the initial value of y when x = 0 p is the percent change (written as a decimal) x is the input variable (very often time) • The equation for the previous example is y = 192 * (1 + (-0.5))x, or y = 192 * .5x

  6. Exponential Growth • If the percentage change p is greater than 0, then we call the relationship exponential growth • If the percentage change p is less than 0, we call the relationship exponential decay • Many exponentials grow (or decrease) very rapidly eventually, but they also can be very, very flat, sometimes deceptively so.

  7. Exponential Growth • If a quantity grows by a fixed percentage change, it grows exponentially • Say a quantity grows by p% each year. After one year, A will become A + Ap, or A*(1+p) • After the second year, you multiply again by 1+p, or A*(1+p)2 • After n years, it is A*(1+p)n (look familiar?)

  8. Exponential Relationships • For example, the US population is growing by about 0.8% each year. In 2000, the population was 282 million. A B C Year Population Population by adding by multiplying percent by growth factor (preferred form) 2000 282 282 2001 =B2+B2*0.008 =C2*(1+.008) 2002 2003

  9. Exponential Relationships • What if a country’s population was decreasing by 0.2% per year? A B C Year Population Population by adding by multiplying percent by growth factor (preferred form) 2000 344 344 2001 =B2-B2*0.002 =C2*(1+(-.002)) 2002 2003

  10. A Very Common Exponential Growth • Every time you buy something you pay sales tax (let’s say its 8.5%) • The item you purchase is $39.00 • Total price = 39 * (1+0.085)1 • Total price = 42.315, or $42.32

  11. Another Example • A bacteria population is at 100 and is growing by 5% per minute • How many bacteria cells are present after one hour (60 minutes)? • You could solve it using a spreadsheet…

  12. Let’s make this chart in Excel.

  13. Another Example • Or you could skip the spreadsheet and solve it mathematically Population = 100 * (1+.05)60 Note: 60 is an exponent in the above equation (but not in the spreadsheet)

  14. What If? • What if you knew the final population and wanted to figure out how long it would take to arrive at this answer? For example, when will the Y population be = 1000? • You could look at the spreadsheet, but… • You might have to make a guess • Let’s not guess, let’s use math

  15. What If? • Let’s use logarithms 1000 = 100(1+.05)X 10 = 1.05X log(10) = log(1.05)X log(10) = x * log(1.05) 1 = x * 0.021 x = 1/0.021 x = 47.61

  16. Is Exponential Modeling Useful? • Populations tend to grow exponentially • When an object cools, the temperature decreases exponentially towards the ambient temp • Radioactive substances (both good and bad) decay exponentially • Money accumulating in a bank at a fixed rate of interest increases exponentially • Viruses and even rumors spread exponentially

  17. Radioisotope Dating • What is radioactivity? • What is it good for? • What is the connection with exponential growth / decay?

  18. Radioisotope Dating • So what is radioisotope dating? • The radioisotope age of a specimen is obtained from a calculation of the time that would be required for unstable parent atoms [P] to spontaneously convert to daughter atoms [D] in sufficient amount to account for the present D/P ratio in the specimen. • Unstable Carbon, Uranium, Potassium often used (Carbon 14 (12), Uranium 238, Potassium 40(39))

  19. Example • The 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% every 100 years • Let’s create a spreadsheet which calculates this exponential delay

  20. Example Years after Death % Carbon Remaining 0 100 100 =B2 * (1 - 0.01202) 200 Stop when % Carbon Remaining = 78% ? Let’s go to the lab.

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