From beginner to expert, IDNLearn.com has answers for everyone. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.
Sagot :
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.
### 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.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.