Question

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

Time (in minutes)
x Air (in ounces)
f(x)
0 64
1 56
2 48
3 40
4 32


Find and interpret the meaning of the x-intercept in this scenario.
a (8, 0); 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 (8, 0); the time it takes to fill up inflatable with air

Answer 1

Answer: Option a.

Explanation:

The equation of the line in Slope-Intercept form is:

`y=mx+b`

Where "m" is the slope and "b" is the y-intercept.

Find the slope of the line with this formula:

`m=(y_2-y_1)/(x_2-x_1)`

Choose two points from the table and substitute the coordinates into the formula. Then, the slope is:

`m=(32-64)/(4-0)=-8`

By definition, the line intersects the y-axis when the value of "x" is zero. So, based on the table, the y-intercept is:

`b=64`

Then the equation of the line is:

`y=-8x+64`

The line intersects the x-axis when the value of "y" is zero, then substituting y=0 into the equation of this line, and solving for "x", you get:

`0=-8x+64\\\\-64=-8x\\\\x=8`

So, the point of intersection with the x-axis is:

`(8,0)`

Since the table shows the amount of air leaking from an inflatable as a function of time, you can conclude that meaning of the x-intercept in this scenario is: "The time it takes all the air to leave the inflatable".