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Exponential Functions Test Part Two

Question 3 (Essay Worth 10 points)

Belinda is thinking about buying a house for [tex]$\$286,000[tex]$[/tex]. The table below shows the projected value of two different houses for three years:

\[
\begin{tabular}{|c|c|c|c|}
\hline
Number of years & 1 & 2 & 3 \\
\hline
House 1 (value in dollars) & 294,580 & $[/tex]303,417.40[tex]$ & $[/tex]312,519.92$ \\
\hline
House 2 (value in dollars) & 295,000 & 304,000 & 313,000 \\
\hline
\end{tabular}
\]

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 [tex]$f(x)[tex]$[/tex], in dollars, after [tex]$[/tex]x$[/tex] years. (4 points)

Part C: Belinda wants to purchase a house that would have the greatest value in 25 years. Will there be any significant difference in the value of either house after 25 years? Explain your answer, and show the value of each house after 25 years. (4 points)

Sagot :

GinnyAnswer
Let's address the problem piece by piece.

### Part A: Determine the Type of Function (Linear or Exponential)

House 1:
1. The initial value is [tex]$286,000, and the values for the next three years are $[/tex]294,580, [tex]$303,417.40, and $[/tex]312,519.92.
2. Calculate the year-over-year growth:
- Year 1 to Year 2: [tex]$\frac{303,417.40}{294,580} \approx 1.030$[/tex]
- Year 2 to Year 3: [tex]$\frac{312,519.92}{303,417.40} \approx 1.030$[/tex]
- Since the growth rate is consistent, the values suggest an exponential function.

House 2:
1. The initial value is [tex]$286,000, and the values for the next three years are $[/tex]295,000, [tex]$304,000, and $[/tex]313,000.
2. Calculate the year-over-year growth:
- Year 1 to Year 2: [tex]$\frac{304,000}{295,000} \approx 1.0305$[/tex]
- Year 2 to Year 3: [tex]$\frac{313,000}{304,000} \approx 1.0296$[/tex]
- Although there's a slight variation, it maintains an approximate linear growth rate. Examining differences:
- Year 1 to Year 2: [tex]$304,000 - 295,000 = 9,000$[/tex]
- Year 2 to Year 3: [tex]$313,000 - 304,000 = 9,000$[/tex]
- Since the differences are consistent, the trend suggests a linear function.

### Part B: Write Functions Describing Each House's Value

House 1:
- The growth rate, calculated approximately as 1.03, showcases exponential growth:
[tex]\[ f_1(x) = 286,000 \times (1.03)^x \][/tex]

House 2:
- The consistent increase of \[tex]$9,000 per year describes linear growth: \[ f_2(x) = 286,000 + 9,000x \] ### Part C: Value of Each House After 25 Years House 1: \[ f_1(25) = 286,000 \times (1.03)^{25} \] Calculate: \[ (1.03)^{25} \approx 2.093 \, \text{(using a calculator)} \] Hence, \[ f_1(25) = 286,000 \times 2.093 \approx 598,598 \] House 2: \[ f_2(25) = 286,000 + 9,000 \times 25 \] Calculate: \[ 9,000 \times 25 = 225,000 \] Hence, \[ f_2(25) = 286,000 + 225,000 = 511,000 \] ### Conclusion Belinda should purchase House 1, as its projected value after 25 years ($[/tex]598,598[tex]$) is higher than that of House 2 ($[/tex]511,000$). The exponential growth of House 1 leads to a significantly higher value in the long term.
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