The following table shows the amount of air leaking from an inflatable as a function of time:

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{l}
Time (in minutes)
\end{tabular} & \begin{tabular}{l}
Air (in ounces) \\
[tex]$f(x)$[/tex]
\end{tabular} \\
\hline 0 & 64 \\
\hline 1 & 56 \\
\hline 2 & 48 \\
\hline 3 & 40 \\
\hline 4 & 32 \\
\hline
\end{tabular}

Find and interpret the meaning of the [tex]$x$[/tex]-intercept in this scenario.

A. [tex]$(\delta, 0)$[/tex], the time it takes all the air to leave the inflatable
B. (64, 0): the time it takes all the air to leave the inflatable
C. (64, 0); the time it takes to fill up inflatable with air
D. [tex]$(6,0)$[/tex], the time it takes to fill up inflatable with air



Answer :

Let's analyze the question step-by-step.

From the table, we can observe that the amount of air decreases uniformly over time. Specifically, every minute, 8 ounces of air leak out of the inflatable. We can express the amount of air as a linear function of time with the following form:

[tex]\[ f(x) = 64 - 8x \][/tex]

Here,
- [tex]\( x \)[/tex] represents the time in minutes,
- [tex]\( f(x) \)[/tex] represents the amount of air remaining in ounces.

We're asked to find and interpret the [tex]\( x \)[/tex]-intercept of this function. The [tex]\( x \)[/tex]-intercept occurs when the amount of air remaining, [tex]\( f(x) \)[/tex], is zero.

To find the [tex]\( x \)[/tex]-intercept:
1. Set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:

[tex]\[ 0 = 64 - 8x \][/tex]

2. Rearrange the equation to isolate [tex]\( x \)[/tex]:

[tex]\[ 8x = 64 \][/tex]

3. Solve for [tex]\( x \)[/tex]:

[tex]\[ x = \frac{64}{8} = 8 \][/tex]

Thus, the [tex]\( x \)[/tex]-intercept is [tex]\( (8, 0) \)[/tex].

Interpretation:
The [tex]\( x \)[/tex]-intercept [tex]\( (8, 0) \)[/tex] tells us that it takes 8 minutes for all the air to completely leak out of the inflatable. This means, after 8 minutes, there will be no air left in the inflatable. Therefore, the correct answer is:

- [tex]\( (8, 0) \)[/tex], the time it takes all the air to leave the inflatable