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## PowerPoint Slideshow about ' PRECALCULUS I' - mardi

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FIVE COMMON TYPES OF MATHEMATICAL MODELS

1. Exponential Growth

2. Exponential Decay

3. Gaussian Model

4. Logistics Growth

5. Logarithmic Model

Find the annual rate (%) for a $10,000

investment to double in 5 years with

continuous compounding.

A = P ert with P = 10000, A = 20000, t = 5

20000 = 10000er(5) or 2 = e5r

ln 2 = ln e5r gives ln 2 = 5r(ln e) = 5r

r = (ln 2)/5 = 0.1386 or r is 13.9%.

The half life of carbon 14 is 5730 years.

Find the equation y = a e bx if a = 3 grams.

0.5(3) = 3 eb(5730) or 0.5 = e5730b

ln 0.5 = ln e5730b gives

ln 0.5 = 5730b(ln e) = 5730b

b = (ln 0.5)/5730 = -0.12097

Thus the equation is y = 3 e -0.12097x .

3. Write the exponential equation of the line that passes through (0,5) and (4,1).

The equation is of the form y = a e bx .

(0,5) yields 5 = a eb(0) or 5 = a e0 or a = 5.

(4,1) yields 1 = 5 eb(4) or 0.2 = eb(4)

ln 0.2 = ln e4b gives ln 0.2 = 4b(ln e) = 4b

b = (ln 0.2)/4 = -0.402359

Thus the equation is y = 5 e -0.402359x .

4. The number of bacteria N is given by the model N = 250 e kt with t in hours.

If N = 280 when t = 10, estimate time for bacteria to double.

The point (10,280) yields 280 = 250 eb(10)

1.12 = e10b or ln 1.12 = ln e10b

ln 1.12 = 10b(ln e) = 10b

b = (ln 1.12)/10 = 0.0113329

Thus the equation is y = 250 e 0.0113329t .

5. The time, t, elapsed since death and the body temperature, T, at room temperature of 70 degrees is given by

If the body temperature at 9:00 a.m. was 85.7 degrees, estimate time of death.

5. If the body temperature at 9:00 a.m. was 85.7 degrees, estimate time of death.

t = -2.5 ln 0.54895 = 1.499 or t = 1.5 hrs

So time of death was 1.5 hrs before 9 a.m Thus the time of death was 7:30 a.m.

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