Who gets this?

Question

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 * a^t , 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.


Type formula above as H = p * a^t

Answer 1

larger. Say the function for H = p * a^t is H= 2^t. At t =1, 2 homes are built. At t =2, 4 homes are built. At t = 3, 8 homes are built. As you can see, as t gets bigger and bigger, The number of homes increases drastically. Say t = 16, then The number of homes built would be a whopping 65536 homes!
Lets say that H = (1/2)^t (Hypothetically). If t = 1, .5 houses are built. At t =2, (1/2)^2 or 1/4 houses are built. As t= 3, 1/8 houses are built. As you see, H gets smaller and smaller

General rule
Given y = p*a^t, where p>0, if a>1, then as t gets larger and larger, so does y
If 0
If you were a home builder, looking for a job, what would you do? If 01, the number of houses each year would increase by a lot. Plenty of workers would be needed each year to complete the task. Thus if you were a home builder, you would hope that a>1!!!.

I hope this helps.

Answer 2

Think about it. If a is less than one, then as t gets larger H gets smaller. If a is 0.5, for example, then after 2 years H= p* 0.5^2 which is H=p*(0.5*0.5), or H=P*0.25. After 3 years H=p*(0.5*0.5*0.5), or H=p*0.125. If you were looking for work you wouldn't go for a place where the number of homes built decreases each year.

If, on the other hand, a is 2, the after 2 years H=p*(2*2), after 3 years H=p*(2*2*2) and so on, with more homes being built every year.

Answer 3

You'd want it greater than 1.

The product of an exponent of a fraction is always less than the original fraction.

Examples:
.75^3 = .421
.25^5 = .000976

Anything greater than one will increase above the original number.

Examples:
2^4 = 16
5.33^3 = 151.42

Answer 4

look at the evolution of H as a function of t:

dH/dt = p*ln(a)*a^t
(a^t = e^tln(a), d(e^tln(a))/dt = ln(a)*e^tln(a))

if 0 if a>1, ln(a)>0 and the number of houses increases with time.

As a homebuidler, you definitely want a>1 to get more and more jobs.

Answer 5

if you were a homebuilder you would not be smart enough to answer that question.

Now if you were an architect, different story.