Question

A painting valued at $25,000 was purchased by an art collector. The function p(t) = 25,000 · (1.09)t models the value of the painting, p(t), in dollars, after a given number of years, t. What is the percent rate of change, and how is it interpreted in the context of the problem?

Answer 1

The percent rate of change in this context is given by the constant factor in the exponential function, which is 0.09 or 9%. \\\\This means that the value of the painting is increasing by 9% each year. In other words, the painting's value is growing at a rate of 9% annually."

Explanation:

Step 1: Start with the given function p(t) = 25,000 · (1.09)t, which models the value of the painting after a given number of years, t.\\\\

step 2: Identify the constant factor in the function, which is the number 1.09. This represents the percent rate of change. In this case, the constant factor is 0.09, which can also be written as 9% in decimal form.\\\\

step 3: Interpret the percent rate of change in the context of the problem. The value of the painting is increasing by 9% each year. This means that the painting's value is growing at a rate of 9% annually.\\\\This means that if the painting is stored for one year, its value will increase by 9% (or $2,250), resulting in a new value of $27,250. Similarly, after two years, the value will increase by another 9% (or $2,452.50), resulting in a new value of $29,702.50, and so on."