Exponential Growth
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This document discusses exponential growth formulas and provides examples of how to model growth using exponential equations. The key aspects are: - The exponential growth formula is y = abx, where a is the starting amount, b is the growth factor (1 + rate), and x is the number of times the growth is applied. - The first example models the increasing daily cost of hospital care from 1985 to 1995 using an 8.6% annual growth rate. - Subsequent examples demonstrate accounting balances growing at 6.5% interest compounded annually, quarterly, semi-annually, and monthly to show the effects of more frequent compounding.
Exponential Growth
- 2. Growth Formula y= a b x a > 0 & b = 1 + rate a is the starting amount b is the base (growth factor) Don’t forget to convert the rate from a % to a decimal BEFORE you add it to the 1 x is the number of increases
- 3. Growth Example 1 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. Write an equation to model the cost of hospital care. Use the equation to find the approximate cost per day in 1995.
- 4. Example 1 Write an equation to model the cost of hospital care. y= a b x a = 460 Rate is 8.6% -- convert to decimal b = 1.086 y= 460 ● 1.086 x
- 5. Example 1 Use the equation to find the approximate cost per day in 1995. y= 460 ● 1.086 x So how many years are there between 1985 and 1995? That’s the x. y = 460 ● 1.086 10 y ≈ 1049.68
- 6. Use your calculator After you find the equation, put it in the Function Editor and use your table of values to find the various increases asked for. Remember to tab over to the Y column so that you can see the complete value
- 7. Example 2 Your parents deposited $500 in an account paying 6.5% interest, compounded annually, when you were born. y= a ● b x a = 500 b = 1.065 y= 500 ● 1.065 x
- 8. Growth Example 2 Find the account balance after 18 years. y= 500 ● 1.065 18 $1553.33
- 9. Example 3 Suppose the amount paid in example 2 paid interest compounded quarterly instead of annually. y= a ● b x a = 500 b = 1 + .065/4 – this is because you the 6.5% is divided out over the year.
- 10. Example 3 Find the account balance after 18 years. y= 500 ● (1 + .065/4) ( 18 ● 4 ) – Change the exponent to 18 x 4 because 4 payments were made per year for the 18 years! $1595.92
- 11. Example 3 Let’s do the same problem compounded semi-annually. y= 500 ● (1 + .065/2) ( 18 ● 2 ) $1581.29 Compounded monthly? y= 500 ● (1 + .065/12) ( 18 ● 12 ) $1605.92