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.
1) Let's say we choose p = 150 and a = 0.8.
2) To find the number of homes built, H, at three different values of time t, let's say t = 1, 3, and 5 years.
For t = 1 year:
H = p * a^t
H = 150 * 0.8^1
H = 120 homes
For t = 3 years:
H = p * a^t
H = 150 * 0.8^3
H = 76.8 homes
For t = 5 years:
H = p * a^t
H = 150 * 0.8^5
H = 49.15 homes
3) In this market, we have chosen p = 150 (initial number of homes built) and a = 0.8 (growth factor).
From our calculations, we can see that after 1 year, the number of homes built would be 120. After 3 years, the number of homes built would be 76.8. And after 5 years, the number of homes built would be 49.15.
Based on these calculations, it seems that the number of new homes being built is decreasing over time. If you were a homebuilder, this suggests that the market is not growing significantly, and the demand for new homes may be decreasing.
In the long run, it may not be favorable to continue building homes in this market, as the demand seems to be declining. It might be more beneficial to explore other markets or industries with higher growth potential.
Sure, let's go through the steps to analyze whether it would be profitable to be a homebuilder in this market.
Step 1: Choose a value for "p"
Let's say we choose p = 150. This represents the initial number of homes built.
Step 2: Choose a value for "a"
Let's choose a = 1.1. This growth factor indicates that the number of homes built will increase by 10% each year.
Now, let's insert these values into the formula H = p * a^t.
1) Inserting the values into the formula:
H = 150 * (1.1)^t
2) Calculating the number of homes built at three different values of time, t:
Let's choose t = 0, t = 1, and t = 5.
At t = 0:
H = 150 * (1.1)^0
H = 150 * 1
H = 150 homes
At t = 1:
H = 150 * (1.1)^1
H = 150 * 1.1
H = 165 homes
At t = 5:
H = 150 * (1.1)^5
H = 150 * 1.61051
H ≈ 241 homes (rounded to the nearest whole number)
3) Summary and analysis:
Based on the calculations, we can see that the number of homes built increases over time. In the first year, there are 150 homes built, and by the fifth year, the number increases to around 241 homes. This growth rate suggests that there is a relatively healthy demand for new homes in this market.
As a homebuilder, if you were to continue building homes in this market, it could be profitable in the long run. The increasing trend indicates a potential for sustained growth and demand for new homes. However, further market research and analysis would be needed, such as considering competition, market saturation, and economic factors, to fully assess the viability of being a homebuilder in this specific market.