The table shows the approximate height of a football after it is kicked. Tell whether the rates of change for successive time intervals are constant or variable

The rates of changes of the heights of a football after it is kicked with respect to the time for this case are 34, 12 and -10 . The rates of changes are variable.

What is a variable and a constant?

A variable is a quantity that changes in its value, but a constant remains fixed.

How to measure the rate of change of something as some other value changes?

Suppose that we have to measure the rate of change of y as x changes, then we have:

[tex]Rate = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]

where we have

[tex]\rm when \: x=x_1, y = y_1\\when\: x = x_2, y= y_2[/tex]

Remember that, we divide by the change in independent variable so that we get some idea of how much the dependent quantity changes as we change the independent quantity by 1 unit.

(5 change per 3 unit can be rewritten as 5/3 change per 1 unit)

For this case, we get the rates as:

For rate between first two values,

[tex]Rate_1 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{17-0}{0.5-0} = 34[/tex]

For rate between second two values,

[tex]Rate_2 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{30-17}{1.5-0.5} = 13[/tex]

For rate between third two values,

[tex]Rate_3 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{25-30}{2-1.5} = -10[/tex]

We see that the rate of change of height of the football considered with respect to time is changing, so its a variable.

Thus, the rates of changes of the heights of a football after it is kicked with respect to the time for this case are 34, 12 and -10 . The rates of changes are variable.

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