Respuesta :
Hello there. To solve this question, we'll have to remember some properties about growth.
Say the initial value P of a painting was increasing at a percentage rate r per year for t years. The final value of the painting after that amount of time will be given by the formula:
[tex]P_f=P_i\cdot(1+r)^t[/tex]In this case, the percentage must be converted to decimals, dividing it by 100, so in general you may have
[tex]P_f=P_i\cdot\left(1+\dfrac{r}{100}\right)^t[/tex]Okay. Now we can solve the question.
We know the initial value of the painting in the year 2021: $490.000
The value has been increasing at the rate of 4% per year. This means that r = 4% and we convert it to decimals in the formula.
We want to calculate its final price after 18 years, that is, when t = 18.
Okay, plugging the values in the formula, we'll get
[tex]\begin{gathered} P_f=490000\cdot\left(1+\dfrac{4}{100}\right)^{18} \\ \\ P_f=490000\cdot(1+0.04)^{18} \\ \\ P_f=490000\cdot1.04^{18} \end{gathered}[/tex]Using a calculator to find an approximation for the power, we'll get
[tex]P_f\approx490000\cdot2.026[/tex]Multiplying the values,
[tex]P_f\approx\$992.740[/tex]This is the approximate value of this painting in 18 years.
