Question

if a painting is worth $135 and increasing by a factor of 1.2 every year for 70 years in the first year its worth $162 and in the second year its worth $194.40

Answer 1

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 \( r = 1.2 \):

1. At the end of the first year \( (t = 1) \):
\[
V(1) = 135 \times 1.2^1 = 135 \times 1.2 = 162
\]

2. 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**.