Let [tex]$g(t)$[/tex] give the market value (in [tex]$1000s$[/tex] of dollars) of a house in year [tex][tex]$t$[/tex][/tex]. What does the statement [tex]\frac{g(3)-g(0)}{3-0}=5[/tex] tell you about the house?

A. The house's market value increased at an average rate of [tex]$\$5000$[/tex] per year between years [tex]$t=0[tex]$[/tex] and [tex]$[/tex]t=3$[/tex].
B. The house's market value decreased at an average rate of [tex]$\[tex]$5000$[/tex][/tex] per year between years [tex]$t=0$[/tex] and [tex]$t=3$[/tex].
C. The house's market value decreased by [tex][tex]$\$[/tex]5000$[/tex] between years [tex]$t=0$[/tex] and [tex]$t=3$[/tex].
D. The house's market value increased by [tex]$\[tex]$5000$[/tex][/tex] between years [tex]$t=0$[/tex] and [tex]$t=3$[/tex].
E. The average house's market value between years [tex][tex]$t=0$[/tex][/tex] and [tex]$t=3$[/tex] was [tex][tex]$\$[/tex]5000$[/tex].



Answer :

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].