To analyze the statement [tex]\(\frac{g(3) - g(0)}{3 - 0} = 5\)[/tex], let's break it down step by step:
1. Understanding the Expression:
- The given expression [tex]\(\frac{g(3) - g(0)}{3 - 0} = 5\)[/tex] is a formula for the average rate of change of the function [tex]\(g(t)\)[/tex] over the interval from [tex]\(t = 0\)[/tex] to [tex]\(t = 3\)[/tex].
- Here, [tex]\(g(t)\)[/tex] represents the market value of the house in thousands of dollars (\[tex]$1000s).
2. Average Rate of Change:
- The average rate of change is calculated as the change in the function value over the change in time: \(\frac{\Delta g}{\Delta t}\).
- Specifically, \(\Delta g = g(3) - g(0)\) and \(\Delta t = 3 - 0\).
- Hence, the equation becomes \(\frac{g(3) - g(0)}{3 - 0} = 5\).
3. Interpreting the Rate:
- The statement tells us that the average rate of change of the market value is 5. But remember, this rate is in units of \$[/tex]1000s per year.
4. Converting Units:
- Since the market value [tex]\(g(t)\)[/tex] is in thousands of dollars, a rate of change of 5 implies an increase of \[tex]$5000 per year (because 5 * \$[/tex]1000 = \[tex]$5000).
5. Conclusion:
- The positive value 5 indicates an increase.
Thus, the statement \(\frac{g(3) - g(0)}{3 - 0} = 5\) tells us that the house's market value increased at an average rate of \$[/tex]5000 per year between years [tex]\(t=0\)[/tex] and [tex]\(t=3\)[/tex].
Here is the concise conclusion:
The house's market value increased at an average rate of \[tex]$5000 per year between years \(t = 0\) and \(t = 3\).
Therefore, the correct answer is:
- The house's market value increased at an average rate of \$[/tex]5000 per year between years [tex]\(t=0\)[/tex] and [tex]\(t=3\)[/tex].
