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Question
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}, (H = p*a^t) where p is the number of new homes built in the first year recorded, and t is the number of years. You are going to decide if you would like to be a homebuilder in this market.
- Step 1 is to choose a value for “p” between 100 and 200; this is the initial number of homes built.
- Step 2 is to choose a value for “a”; this is the growth factor – you can choose “a” to be any number between 0 and 1 “OR” choose “a” to be any number greater than 1.
NB: Do not choose 0 or 1, as these are trivial cases.
1) Insert the chosen values for “p” and “a” into the formula listed above.
2) Use the formula to find the number of homes built, H, at any three values of time, t, in years that you want.
3) Show your calculations and put units on your final answer!
4) Provide a written summary of your results explaining them in the context of the original problem. If you were a homebuilder, would you be interested in continuing to build homes in this market over the long run? Explain why or why
Solution:
Take the initial value of “p” to be 125 in this market which is gradually turning to be a commercial center. Now, the growth rate, “a” is determined by a number of factors (Stapel, 2010). In a depreciating economy, the value of ‘a’ is expected to be less than one. In this case, the value of ‘a’ will be taken as 1.1.
Therefore,
H = p * a^{t }but p = 125 and a = 1.1
Hence H = 125 *1.1^{t . }Taking the values of t as 5, 10, 20 years, and inserting them into the expression, the results are as tabulated below.
Time, t (years) |
1 |
5 |
10 |
20 |
No. of houses, H |
125 |
201 |
324 |
841 |
Results Summary
A mathematical projection on the trend of home building is a factor of a number of variables which when combined together in an equal proportion will determine the growth rate (Goldstein & Schneider, 2006). The growth rate in whatever case can either be a value between 0 and 1 indicating of a decreasing trend in what is being anticipated or have a value above 1 indicative of an increasing trend.
These kinds of projections are founded on exponentially varying factors as time goes by. In this research, the number of houses built, H over a period of time, t is entirely depended on other factors which cannot be put in this expression; otherwise it would become too complex. These other factors are summarized with the use of the value “a” herein referred to as the growth rate.
Decision Making
Such mathematical models as in this case are very useful while making decision on the feasibility of any project (Stapel, 2010). Looking at the trend followed in this case, I would confidently settle on this project of a home building. This is due to the reason that the factors summarized by the value “a” are apparently favorable for its success.
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