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Answer:
The average rate of change of the height over the interval [1, 2.5] is - 56 feet per second.
Step-by-step explanation:
Let [tex]s(t) = -16\cdot t^{2} + 150[/tex]. Geometrically speaking, average rate of change over a given interval ([tex]\bar v[/tex]), measured in feet per second, is determined by definition of secant line, which is defined:
[tex]\bar v = \frac{s(t_{2})-s(t_{1}) }{t_{2}-t_{1}}[/tex] (1)
Where:
[tex]s(t_{1})[/tex], [tex]s(t_{2})[/tex] - Initial and final position of the object, measured in feet.
[tex]t_{1}[/tex], [tex]t_{2}[/tex] - Initial and final times, measured in seconds.
If we know that [tex]t_{1} = 1\,s[/tex] and [tex]t_{2} = 2.5\,s[/tex], then the average rate of change over the interval [tex][1,2.5][/tex]:
[tex]s(1) = -16\cdot (1)^{2}+150[/tex]
[tex]s(1) = 134[/tex]
[tex]s(2.5) = -16\cdot (2.5)^{2}+150[/tex]
[tex]s(2.5) = 50[/tex]
[tex]\bar v = \frac{50\,ft-134\,ft}{2.5\,s-1\,s}[/tex]
[tex]\bar v = -56\,\frac{ft}{s}[/tex]
The average rate of change of the height over the interval [1, 2.5] is - 56 feet per second.