8-1 Exponential Growth & Decay

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8-1 Exponential Growth & Decay

  1. 1. Exponential Growth & Decay
  2. 2. Growth Formula <ul><li>y= a b x </li></ul><ul><li>a > 0 & b = 1 + rate </li></ul><ul><li>a is the starting amount </li></ul><ul><li>b is the base (growth factor) </li></ul><ul><li>Don’t forget to convert the rate from a % to a decimal </li></ul><ul><li>x is the number of increases </li></ul>
  3. 3. Growth Example 1 <ul><li>Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8.6% per year. In 1985, hospital costs were an average of $460 per day. </li></ul><ul><li>Write an equation to model the cost of hospital care. </li></ul><ul><li>Use the equation to find the approximate cost per day in 1995. </li></ul>
  4. 4. Growth Example 1 <ul><li>Write an equation to model the cost of hospital care. </li></ul><ul><li>y= a b x </li></ul><ul><li>a = 460 </li></ul><ul><li>Rate is 8.6% -- convert to decimal </li></ul><ul><li>b = 1.086 </li></ul><ul><li>y= 460 ● 1.086 x </li></ul>
  5. 5. Growth Example 1 <ul><li>Use the equation to find the approximate cost per day in 1995. </li></ul><ul><li>y= 460 ● 1.086 x </li></ul><ul><li>So how many years are there between 1985 and 1985? </li></ul><ul><li>That’s the x. </li></ul><ul><li>y= 460 ● 1.086 10 </li></ul><ul><li>y ≈ 1049.68 </li></ul>
  6. 6. Growth Example 2 <ul><li>Your parents deposited $500 in an account paying 6.5% interest, compounded annually, when you were born. </li></ul><ul><li>y= a ● b x </li></ul><ul><li>a = 500 </li></ul><ul><li>b = 1.065 </li></ul><ul><li>y= 500 ● 1.065 x </li></ul>
  7. 7. Growth Example 2 <ul><li>Find the account balance after 18 years. </li></ul><ul><li>y= 500 ● 1.065 18 </li></ul><ul><li>$1553.33 </li></ul>
  8. 8. Growth Example 3 <ul><li>Suppose the amount paid in example 2 paid interest compounded quarterly instead of annually. </li></ul><ul><li>y= a ● b x </li></ul><ul><li>a = 500 </li></ul><ul><li>b = 1 + .065/4 </li></ul><ul><li>y= 500 ● 1.01625 x </li></ul>
  9. 9. Growth Example 3 <ul><li>Find the account balance after 18 years. </li></ul><ul><li>y= 500 ● 1.01625 ( 18●4 ) </li></ul><ul><li>$1610.83 </li></ul>
  10. 10. Decay Formula <ul><li>y= a b x </li></ul><ul><li>a > 0 & b = 1 - rate </li></ul><ul><li>a is the starting amount </li></ul><ul><li>b is the base (decay factor) </li></ul><ul><li>Don’t forget to convert the rate from a % to a decimal </li></ul><ul><li>x is the number of decreases </li></ul>
  11. 11. Decay Example 1 <ul><li>To treat some forms of cancer, doctors use radioactive iodine. Use the graph on page 373 to find out how much iodine is left in the patient 8 days after a patient receives a dose of 20mCi (millicuries). </li></ul><ul><li>Since the x value represents the days, the y value represents the amount of iodine left -> 10. </li></ul><ul><li>How much is left after 24 days? </li></ul><ul><li>About 2.5 </li></ul>
  12. 12. Decay Example 2 <ul><li>An exponential function models the amount of whole milk each person in the United States drinks in a year. Graph the function y=21.5 ● 0.955 x , where y is the number of gallons of whole milk and x is the number of years since 1975. </li></ul><ul><li>What procedures will you use to graph this function? </li></ul>
  13. 13. Decay Example 3 <ul><li>Use the equation y=21.5 ● 0.955 x to find the annual percent of decrease in whole milk consumption in the United States. </li></ul><ul><li>.955 = 95.5% </li></ul><ul><li>100% - 95.5% = 4.5% decrease per year </li></ul>

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