The volume of helium inside the balloon decreased at a rate of 0.1 cubic feet per minute. The linear equation that represents the given data is y = -0.1x + 6 and its slope m = -0.1.
What is the equation of a line passing through two points?
The equation of the line passing through the points (x1, y1) and (x2, y2) is
(y - y1) = m(x - x1).
Where m is the slope of the line and is calculated by
m = (y2 - y1)/(x2 - x1)
Calculation:
The given data in the table represents the volume of helium (cu ft), inside a balloon relative to the elapsed time in minutes.
x: 5, 10, 15, 20, 25, 30, 35, 40
y: 5.5, 5, 4.5, 4, 3.5, 3, 2.5, 2
From this table, consider two points (5, 5.5) and (10, 5)
So, the equation of the line is
y- y1 = m(x - x1)
Where slope m = (y2 - y1)/(x2 - x1)
⇒ m = (5 - 5.5)/(10 - 5)
⇒ m = -0.5/5
∴ m = -0.1
Thus, the equation is
y - 5.5 = -0.1(x - 5)
⇒ y - 5.5 = -0.1x + 0.5
⇒ y = -0.1x + 0.5 + 5.5
⇒ y = -0.1x + 6.0
∴ y = -0.1x + 6
Since the slope of the line is negative, the volume of the helium inside the balloon decreased at a rate of 0.1 cubic ft per minute.
So, option 2 is correct.
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