Let's analyze the given problem step by step.
A. Find \( h(t) \)
The function given is:
\[ h(t) = -16t^2 + 80t + 30 \]
(Note: I corrected the function to have a negative coefficient for \( t^2 \) since projectile motion generally has a negative sign reflecting gravity's effect on the height of the object over time.)
B. What does \( h(2.5) \) represent in the context of this problem?
To find \( h(2.5) \):
\[ h(2.5) = -16(2.5)^2 + 80(2.5) + 30 \]
Calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Now compute \( h(2.5) \):
\[ h(2.5) = -16(6.25) + 80(2.5) + 30 \] \[ = -100 + 200 + 30 = 130 \]
Thus, \( h(2.5) = 130 \).
In the context of the problem, \( h(2.5) = 130 \) represents the height of the rock, in inches, at \( t = 2.5 \) seconds after it has been thrown.
C. Does it make sense to find \( h(-10) \) in the context of this problem?
In the context of this problem, \( h(t) \) calculates the height of the rock after it has been thrown. The variable \( t \) represents time in seconds since the rock was thrown, which means \( t \) cannot be negative.
Therefore, finding \( h(-10) \) does not make sense in the context of this problem because it would imply looking for the height of the rock 10 seconds before it was thrown, which is outside the physical context of the scenario. Hence, \( h(-10) \) is not a valid calculation in this case.
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