To find the average rate of change of the value of Painting A over a 3-year period, we need to calculate the change in value over that period and divide by the length of the period.
The value of Painting A after x years is given by the function f(x) = 50,000 (1.064)^x. Therefore, the value after 3 years is f(3) = 50,000 (1.064)^3.
The value of Painting A at the beginning of the 3-year period is f(0) = 50,000 (1.064)^0 = 50,000.
The change in value over the 3-year period is therefore:
f(3) - f(0) = 50,000 (1.064)^3 - 50,000
Simplifying this expression, we get:
f(3) - f(0) = 50,000 (1.064)^3 - 50,000
= 50,000 (1.064^3 - 1)
Using a calculator, we can evaluate this expression to be approximately $8,490.52.
Therefore, the average rate of change of the value of Painting A over the 3-year period is:
Average rate of change = change in value / length of period
= $8,490.52 / 3 years
= $2,830.17 per year
So the average rate of change of the value per year, over a 3 year period, is $2,830.17.