If a painting's value is increasing by a factor of 1.2 each year, we can represent its value in any year as follows:
\[ V(t) = V_0 \times r^t \]
Where:
- \( V(t) \) is the value at year \( t \),
- \( V_0 \) is the initial value (at year 0),
- \( r \) is the growth factor (1.2 in this case),
- \( t \) is the number of years since the start.
Given that the starting value \( V_0 = 135 \) and it grows by a factor of \( r = 1.2 \):
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At the end of the first year \( (t = 1) \): \[ V(1) = 135 \times 1.2^1 = 135 \times 1.2 = 162 \]
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At the end of the second year \( (t = 2) \): \[ V(2) = 135 \times 1.2^2 = 135 \times 1.44 = 194.40 \]
This matches your statement about the painting's value for the first two years, confirming the formula is correct.
To find the value of the painting after 70 years \( (t = 70) \): \[ V(70) = 135 \times 1.2^{70} \]
Using a calculator (or a computational tool) to find \( 1.2^{70} \):
\[ 1.2^{70} \approx 133.062 \]
Now calculate:
\[ V(70) \approx 135 \times 133.062 \approx 17,973.37 \]
So, the estimated worth of the painting after 70 years would be approximately $17,973.37.
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