Library Research Assignment
Please help with Discussion board problem
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 , (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. 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. Show your calculations and put units on your final answer!
3) 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 not.
To complete the discussion board problem, follow these steps:
Step 1: Choose a value for "p" between 100 and 200. Let's say we choose p = 150.
Step 2: Choose a value for "a". We have the option to choose "a" to be any number between 0 and 1 OR any number greater than 1. For this example, let's choose a = 1.5.
Now, let's plug these values into the exponential function H = p * a^t.
1) Inserting the values for "p" and "a" into the formula:
H = 150 * (1.5)^t
2) Use the formula to find the number of homes built, H, at three different values of time, t, in years.
Let's choose t = 2, t = 4, and t = 6.
For t = 2:
H = 150 * (1.5)^2
H = 150 * 2.25
H = 337.5 homes
For t = 4:
H = 150 * (1.5)^4
H = 150 * 5.06
H = 759 homes
For t = 6:
H = 150 * (1.5)^6
H = 150 * 11.39
H = 1708.5 homes
3) 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 not.
Based on the calculations, we have found the number of homes built, H, at different time intervals. At t = 2 years, there are approximately 337.5 homes built. At t = 4 years, there are approximately 759 homes built. At t = 6 years, there are approximately 1708.5 homes built.
As a homebuilder, the key factor to consider is the growth trend represented by the value of "a." In this case, with an "a" value of 1.5, we can observe exponential growth in the number of homes being built over time.
Looking at the results, we can see that the number of homes being built is increasing significantly. If we continue to build homes in this market, we can expect a steady increase in demand. Therefore, it seems like a promising market for a homebuilder in the long run.
Remember to provide further justifications or explanations based on the specific context of the problem and market conditions, as requested by the assignment.