Let's analyze the problem step by step to determine which expression correctly represents the painting's value after 7 years.
1. Initial Value of the Painting: Javier bought the painting for \[tex]$150.
2. Annual Increase Factor: Each year, the value of the painting increases by a factor of 1.15. This means each year, the new value of the painting is 115% of its value from the previous year.
3. Time Period: We are considering a growth period of 7 years.
Now, let's translate this into an expression to find the painting's value after 7 years.
Understanding Exponential Growth:
Each successive year, the painting's value multiplies by 1.15. To model this process, we use exponential growth, where the value after a certain number of years \( t \) can be represented as:
\[ \text{Value after } t \text{ years} = \text{Initial Value} \times (\text{Growth Factor})^t \]
Plugging in the given values:
- Initial Value = \$[/tex]150
- Growth Factor = 1.15
- Time (years) [tex]\( t = 7 \)[/tex]
The correct expression becomes:
[tex]\[ \text{Value after 7 years} = 150 \times (1.15)^7 \][/tex]
Comparing the Options:
Now let's look at each option and see which matches our derived expression:
- (A) [tex]\((150 \cdot 1.15)^7\)[/tex]: This expression mistakenly multiplies [tex]\(150\)[/tex] by [tex]\(1.15\)[/tex] first, and then raises the result to the 7th power, which is incorrect.
- (B) [tex]\(150 + 1.15 \cdot 7\)[/tex]: This expression adds a simplistic linear increase, which does not account for compounding and is therefore incorrect.
- (C) [tex]\(150 \cdot 1.15^7\)[/tex]: This expression correctly denotes the initial value multiplied by the growth factor raised to the 7th power, which matches our exponential growth model.
- (D) [tex]\((150 + 1.15) \cdot 7\)[/tex]: This expression adds [tex]\(150\)[/tex] to [tex]\(1.15\)[/tex] first and then multiplies by [tex]\(7\)[/tex], which is incorrect and does not reflect exponential growth.
Hence, the correct answer is:
(C) [tex]\(150 \cdot 1.15^7\)[/tex]
To verify, if we compute this value, it accurately comes out to approximately \$399.00, confirming that the expression and our interpretation are correct.
