8.3: When good models go bad
When using mathematical models to predict future behavior, it is important to keep in mind that very few trends will continue indefinitely.
Suppose a four year old boy is currently 39 inches tall, and you are told to expect him to grow 2.5 inches a year.
Solution
We can set up a growth model, with \(n = 0\) corresponding to 4 years old.
Recursive form:
\(P_{0}=39\)
\(P_{n}=P_{n-1}+2.5\)
Explicit form:
\(P_{n}=39+2.5 n\)
So at 6 years old, we would expect him to be
\(P_{2}=39+2.5(2)=44\) inches tall
Any mathematical model will break down eventually. Certainly, we shouldn’t expect this boy to continue to grow at the same rate all his life. If he did, at age 50 he would be
\(P_{46}=39+2.5(46)=154\) inches tall \(=12.8\) feet tall!
When using any mathematical model, we have to consider which inputs are reasonable to use. Whenever we extrapolate, or make predictions into the future, we are assuming the model will continue to be valid.