Part 1: Suppose that the number of new homes built, H, in a city over a period of time, t, is graphed on a rectangular coordinate system where time is on the horizontal axis. Suppose that the number of homes built can be modeled by an exponential function, H= p * at , where p is the number of new homes built in the first year recorded. If you were a homebuilder looking for work, would you prefer that the value of a to be between 0 and 1 or larger than 1? Explain your reasoning.
Typing hint: Type formula above as H = p * a^t
Umm. Why the hint? Is that not supposed to be the model eponential function?
Originally Posted by harry
The given model, H = p*at, is not exponential.
Say, H = p * a^t -----(i)
If you are looking for work, you need more new houses to be built. So you need H to increase as time t goes on.
If 0 < a < 1, or "a" is a fraction less than 1, then as t increases, the a^t decreases in value. So H decreases too. No good for you.
Example, a = 0.3
at t=1, a^t = (0.3)^1 = 0.3
at t=2, a^t = (0.3)^2 = 0.09 <----less than 0.3
at t=3, a^t = (0.3)^3 = 0.027 <---less than 0.09
If "a" is greater than 1, then the a^t increases as t increases. So the H increases too. Good for you.
Example, a = 3
at t=1, a^t = (3)^1 = 3
at t=2, a^t = (3)^2 = 9 <----more than 3
at t=3, a^t = (3)^3 = 27 <---more than 9
So which one you prefer?