www.ck12.org
Chapter 6. Transcendental Functions
A = Pert 30, 000 = 10, 000e(0.139)(t) 3 = e0.139t ln 3 = 0.139t ln 3 t= 0.139 = 7.9 years.
Other Exponential Models and Examples Not all exponential growths and decays are modeled in the natural base e or by y = 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 −
11, 000 , 2.2 + e−0.0003x
where p is the price per unit and x is 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?
11, 000 2.2 + e0.0003x 11, 000 = 12, 400 − 2.2 + e0.0003(7000) 11, 000 = 12, 400 − 2.2 + 0.122 = 7, 663.
p = 12, 400 −
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=
4500 , x ≥ 0, 1 + 4499e−0.8x
where P represents the total number of infected students and x is 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). Substituting x = 7 into the model, 309