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? The percent rate of change is 9%, which means the value of the painting increases by 9% per year. The percent rate of change is 9%, which means the value of the painting decreases by 9% per year. The percent rate of change is 91%, which means the value of the painting increases by 9% per year. The percent rate of change is 91%, which means the value of the painting decreases by 9% per year.

Answer 1

A) The percent rate of change is 9%, which means the value of the painting increases by 9% per year.

Explanation:

Given function:

`p(t) = 25000 \cdot (1.09)^t`

The given function p(t) represents the value of the painting in dollars after t years. This is an exponential growth function, where the value of the painting is increasing over time.

The general form of an exponential growth function is given by:

`\large\boxed{f(t)=a(1+r)^t}`

where:

• a is the initial value.
• r is the growth rate per time period (as a decimal).
• t is time.

Comparing the general form to the given function:

• The initial value (a) is $25,000.
• The growth rate (r) is r = 1.09 - 1 = 0.09 = 9%.

Therefore, the percent rate of change is 9%, which means the value of the painting increases by 9% per year.