Question

(02.03 hc)
the following table shows the amount of air leaking from an inflatable as a function of time:
find and interpret the meaning of the x - intercept in this scenario.
(8, 0); the time it takes all the air to leave the inflatable
(64, 0); the time it takes all the air to leave the inflatable
(64, 0); the time it takes to fill up inflatable with air
(8, 0); the time it takes to fill up inflatable with air

Explanation:

Step1: Determine the function type

From the table, we see that as \(x\) (time) increases by \(1\), \(f(x)\) (air) decreases by \(8\). So it is a linear function. The general form of a linear function is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Here, \(b=f(0) = 64\), and \(m=\frac{56 - 64}{1-0}=- 8\). So the function is \(f(x)=-8x + 64\).

Step2: Find the \(x\)-intercept

The \(x\)-intercept is found by setting \(y = f(x)=0\). So we solve the equation \(0=-8x + 64\).
Add \(8x\) to both sides: \(8x=64\).
Divide both sides by \(8\): \(x = 8\).
The \(x\)-intercept is \((8,0)\). In the context of the problem, \(x\) represents time and \(y = f(x)\) represents the amount of air. When \(y = 0\), it means all the air has left the inflatable.

Answer:

\((8,0)\): the time it takes all the air to leave the inflatable.