Let's analyze the function \( h(t) = -16t^2 + 35t + 80 \) which represents the height of the rock \( t \) seconds after it has been thrown.
1. Find \( h(2) \)
To find \( h(2) \):
\[ h(2) = -16(2)^2 + 35(2) + 80 \] \[ = -16(4) + 70 + 80 \] \[ = -64 + 70 + 80 \] \[ = 6 + 80 \] \[ = 86 \]
So, \( h(2) = 86 \).
2. What does \( h(2) \) represent in the context of this problem?
In this context, \( h(2) = 86 \) represents the height of the rock above the ground, measured in feet, 2 seconds after it has been thrown. Therefore, at 2 seconds, the rock is 86 feet above the ground.
3. Does it make sense to find \( h(-4) \) in the context of this problem?
To find \( h(-4) \):
\[ h(-4) = -16(-4)^2 + 35(-4) + 80 \] \[ = -16(16) - 140 + 80 \] \[ = -256 - 140 + 80 \] \[ = -256 - 60 \] \[ = -316 \]
While we can mathematically compute \( h(-4) \), it does not make sense in the context of this problem. The negative time value of \( -4 \) seconds would imply a time 4 seconds before the rock was thrown, which is outside the relevant time frame of the event (the throw). Therefore, finding \( h(-4) \) is not meaningful in this scenario.