RATIONAL FUNCTIONS

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PRECALCULUS I
EXPONENTIAL LOG MODELS
Dr. Claude S. MooreDanville Community College
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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
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1. EXPONENTIAL GROWTH
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.
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2. EXPONENTIAL DECAY
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 .
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EXPONENTIAL EQUATION
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 .
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BACTERIA GROWTH
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 .
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TIME OF DEATH
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 900 a.m. was 85.7
degrees, estimate time of death.
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TIME OF DEATH concluded
5. If the body temperature at 900 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 730 a.m.
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Study and Learn before time runs OUT.