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A computer had 2 gigabytes of data stored on it when Jackie bought it, and she is storing an additional 3.5 gigabytes per year, as shown in the table below. Which of these statements is correct about the linear function that represents this situation?

\begin{tabular}{|c|c|c|c|c|c|}
\hline \multicolumn{6}{|c|}{ Amount of Data Stored on Jackie's Computer } \\
\hline Time (year) & 0 & 1 & 2 & 3 & 4 \\
\hline Gigabytes of Data & 2 & [tex]$2+3.5=5.5$[/tex] & [tex]$5.5+3.5=9$[/tex] & [tex]$9+3.5=12.5$[/tex] & [tex]$12.5+3.5=16$[/tex] \\
\hline
\end{tabular}

A. The initial value represents the 2 gigabytes of data stored on the computer when Jackie bought it and the rate of change represents the 3.5 gigabytes per year that Jackie is storing.

B. The initial value represents the 2 gigabytes per year that Jackie is storing, and the rate of change represents the 3.5 gigabytes of data stored on the computer when Jackie bought it.

C. The initial value represents the 3.5 gigabytes of data stored on the computer when Jackie bought it, and the rate of change represents the 2 gigabytes per year that Jackie is storing.

D. The initial value represents the 3.5 gigabytes per year that Jackie is storing, and the rate of change represents the 2 gigabytes of data stored on the computer when Jackie bought it.



Answer :

GinnyAnswer
To solve this problem, we need to carefully examine the information given and compare it with the possible statements provided to determine which one correctly describes the linear function that represents the situation.

Here are the key details from the problem:

- Initially, the computer had 2 gigabytes of data stored when Jackie bought it.
- Jackie is storing an additional 3.5 gigabytes of data each year.

We need to determine the correct statement about the linear function by analyzing the terms "initial value" and "rate of change."

1. Initial Value: This represents the starting amount of data on the computer when Jackie bought it, which is 2 gigabytes.

2. Rate of Change: This represents the constant amount of additional data Jackie is storing each year, which is 3.5 gigabytes per year.

From the tabulated data we can confirm the yearly increments:

- Year 0: 2 gigabytes (initial value)
- Year 1: 2 + 3.5 = 5.5 gigabytes
- Year 2: 5.5 + 3.5 = 9.0 gigabytes
- Year 3: 9.0 + 3.5 = 12.5 gigabytes
- Year 4: 12.5 + 3.5 = 16 gigabytes

These values align perfectly with the linear model [tex]\( D(t) = 2 + 3.5t \)[/tex], where [tex]\( D(t) \)[/tex] represents the data stored after [tex]\( t \)[/tex] years.

Given the possible statements and the information provided, we can deduce:

- The first option states: "The initial value represents the 2 gigabytes of data stored on the computer when Jackie bought it and the rate of change represents the 3.5 gigabytes per year that Jackie is storing."
- This is correct because the initial value is indeed 2 gigabytes, and the rate of change is 3.5 gigabytes per year.

- The other statements are incorrect for the reasons below:
- The second option: It wrongly states the initial value as the rate of data storage per year and the rate of change as the initial amount of data stored, which is incorrect.
- The third option: It inverts the initial value and rate of change, which is incorrect.
- The fourth option: It makes the same mistake as the third by inverting the values.

Thus, the correct statement is:
The initial value represents the 2 gigabytes of data stored on the computer when Jackie bought it and the rate of change represents the 3.5 gigabytes per year that Jackie is storing.