6.4. Exponential Growth and Decay http://www.ck12.org
A=Pert
30 , 000 = 10 , 000 e(^0.^139 )(t)
3 =e^0.^139 t
ln 3= 0. 139 t
t= 0 ln 3. 139
= 7 .9 years.Other Exponential Models and Examples
Not all exponential growths and decays are modeled in the natural baseeor byy=Cekt.Actually, in everyday life
most are constructed from empirical data and regression techniques. For example, in the business world the demand
function for a product may be described by the formula
p= 12 , 400 − 2. 211 +e,−^0000. 0003 x,wherepis the price per unit andxis the number of units produced. So if the business is interested in basing the price
of its unit on the number that it is projecting to sell, this formula becomes very helpful.
If a motorcycle factory is projecting to sell 7000 units in one month, what price should the factory set on each
motorcycle?
p= 12 , 400 − 2. 211 +,e^0000. 0003 x
= 12 , 400 − 2. 2 +^11 e 0 ,. 0003000 ( 7000 )
= 12 , 400 − 2. 211 +,^0000. 122
= 7 , 663.Thus the factory’s base price for each motorcycle should be set at $7663.
As another example, let’s say a medical researcher is studying the spread of the flu virus through a certain campus
during the winter months. Let’s assume that the model for the spread is described by
P= 1 + 44994500 e− 0. 8 x,x≥ 0 ,wherePrepresents the total number of infected students andxis the time, measured in days. Suppose the researcher
is interested in the number of students who will be infected in the next week (7 days). Substitutingx=7 into the
model,