Answer :
To determine which equation fits the situation described in the problem, let's follow these steps:
1. Identify the Initial Value: Luca originally bought the home for [tex]$115,000. This is the starting value or initial investment.
2. Understand the Rate of Increase: The property value increases by 5% each year. So, each year, the property retains its previous value and gains an additional 5%.
3. Determine the Growth Factor: When property increases by a percentage, we can represent this with a growth factor. A 5% increase means you multiply by 1.05 (where 1 represents the original 100%, and 0.05 represents the additional 5%).
4. Write the Exponential Growth Formula: Exponential growth can be written in the form \( y = a \cdot (1 + r)^x \), where:
- \( y \) is the final amount after \( x \) years,
- \( a \) is the initial amount ($[/tex]115,000),
- [tex]\( r \)[/tex] is the growth rate (0.05 for 5% increase),
- [tex]\( x \)[/tex] is the number of years.
5. Substitute the Values into the Formula: Using the values from our problem:
- Initial Value [tex]\( a = 115,000 \)[/tex],
- Growth Rate [tex]\( r = 0.05 \)[/tex].
Plug these into the formula:
[tex]\[
y = 115,000 \cdot (1.05)^x
\][/tex]
From the choices provided, the correct equation that matches this formula is:
[tex]\[ y = 115,000 \cdot (1.05)^x \][/tex]
Thus, the correct answer is:
[tex]\[ y = 115,000(1.05)^x \][/tex]
This equation correctly reflects the 5% annual increase in property values.
1. Identify the Initial Value: Luca originally bought the home for [tex]$115,000. This is the starting value or initial investment.
2. Understand the Rate of Increase: The property value increases by 5% each year. So, each year, the property retains its previous value and gains an additional 5%.
3. Determine the Growth Factor: When property increases by a percentage, we can represent this with a growth factor. A 5% increase means you multiply by 1.05 (where 1 represents the original 100%, and 0.05 represents the additional 5%).
4. Write the Exponential Growth Formula: Exponential growth can be written in the form \( y = a \cdot (1 + r)^x \), where:
- \( y \) is the final amount after \( x \) years,
- \( a \) is the initial amount ($[/tex]115,000),
- [tex]\( r \)[/tex] is the growth rate (0.05 for 5% increase),
- [tex]\( x \)[/tex] is the number of years.
5. Substitute the Values into the Formula: Using the values from our problem:
- Initial Value [tex]\( a = 115,000 \)[/tex],
- Growth Rate [tex]\( r = 0.05 \)[/tex].
Plug these into the formula:
[tex]\[
y = 115,000 \cdot (1.05)^x
\][/tex]
From the choices provided, the correct equation that matches this formula is:
[tex]\[ y = 115,000 \cdot (1.05)^x \][/tex]
Thus, the correct answer is:
[tex]\[ y = 115,000(1.05)^x \][/tex]
This equation correctly reflects the 5% annual increase in property values.
