John is thinking about buying a house for $179,000. The table below shows the projected value of two different houses for three years.
Number of years 1 2 3
House 1 (value in dollars) 186,160 193,606.40 201,350.66
House 2 (value in dollars) 190,000 201,000 212,000
Part A: What type of function, linear or exponential, can be used to describe the value of each of the houses after a fixed number of years? Explain your answer.
Part B: Write one function for each house to describe the value of the house f(x), in dollars, after x years.
Part C: John wants to purchase a house that would have the greatest value in 30 years. Will there be any significant difference in the value of either house after 30 years? Explain your answer, and show the value of each house after 30 years.
Thanks for the help!
Part A: To determine whether the function that describes the value of the houses is linear or exponential, we need to analyze the pattern in the given values.
For House 1:
The difference between the values at each step appears to be increasing, suggesting an exponential pattern. This can be confirmed by calculating the ratios between consecutive values. Let's look at the ratios for House 1:
193,606.40/186,160 ≈ 1.040 - 1st and 2nd years
201,350.66/193,606.40 ≈ 1.040 - 2nd and 3rd years
The ratios are approximately the same, which indicates exponential growth.
For House 2:
Similarly, let's calculate the ratios between consecutive values for House 2:
201,000/190,000 ≈ 1.058 - 1st and 2nd years
212,000/201,000 ≈ 1.055 - 2nd and 3rd years
Though the ratios are slightly different, they are still relatively close, suggesting exponential growth for House 2 as well.
Therefore, both houses exhibit an exponential growth pattern.
Part B: To write the functions describing the value of each house after x years, we need to identify the common ratio (r) of growth for each house.
For House 1:
The common ratio (r) can be calculated using the ratio between any two consecutive values. Let's use the ratio between the first and second years: r = 193,606.40/186,160 ≈ 1.040.
Hence, the function for House 1 after x years is f(x) = 179,000 * (1.040)^x.
For House 2:
Similarly, the common ratio (r) can be calculated using the ratio between any two consecutive values. Let's use the ratio between the first and second years: r = 201,000/190,000 ≈ 1.058.
Therefore, the function for House 2 after x years is f(x) = 179,000 * (1.058)^x.
Part C: To determine whether there will be a significant difference in the value of either house after 30 years, we can calculate the value of each house at that time.
For House 1:
f(30) = 179,000 * (1.040)^30 ≈ $639,231.87
For House 2:
f(30) = 179,000 * (1.058)^30 ≈ $718,491.56
We can observe that House 2 will have a higher value after 30 years, with an approximate value of $718,491.56, compared to House 1 which will have an approximate value of $639,231.87. Therefore, there will be a significant difference in the values of the two houses after 30 years.
Part B: Write one function for each house to describe the value of the house f(x), in dollars, after x years.
I have f(x) = 110,000(x) as a function for House 2 but what about House 1? Since House 1 progresses at an exponential rate I don't know what to do.
Also, for Part C, obviously I know theirs going to be an increase what I need help with was the value of each house after 30 years.
A. Is the amount of change each year constant? Or does the actual change keep increasing? Linear means a constant change rate.
B. should not be too hard
C. Clearly there will be a major difference, if the table's trend continues.