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
