(a) The selling price of the necklace increases by 5% each year. This can be modeled by the formula for compound interest:
\(P_n = 36000 \times (1.05)^n\)
To find the selling price in 2008, we calculate for \(n = 8\) (since 2008 is 8 years after 2000):
\(P_8 = 36000 \times (1.05)^8\)
Calculating this gives:
\(P_8 = 36000 \times 1.477455 \approx 53200\)
So, the selling price in 2008 is $53,200.
(b) The total amount of money obtained over 10 years is the sum of a geometric series where the first term \(a = 36000\) and the common ratio \(r = 1.05\).
The sum of the first \(n\) terms of a geometric series is given by:
\(S_n = a \frac{r^n - 1}{r - 1}\)
For \(n = 10\):
\(S_{10} = 36000 \frac{(1.05)^{10} - 1}{1.05 - 1}\)
Calculating this gives:
\(S_{10} = 36000 \frac{1.628895 - 1}{0.05}\)
\(S_{10} = 36000 \times 12.5779 \approx 453000\)
So, the total amount of money obtained in the ten-year period is $453,000.