To answer your questions, let's break down each part:
Given Function:
The height of the rock as a function of time is given by: \[ h(t) = -16t^2 + 80t + 30 \] This is a quadratic function representing the height in inches of the rock at time \( t \) seconds after being thrown.
a) Find \( h(2.5) \):
Substituting \( t = 2.5 \) into the height function: \[ h(2.5) = -16(2.5)^2 + 80(2.5) + 30 \] Calculating each term:
- \( (2.5)^2 = 6.25 \)
- \( -16(6.25) = -100 \)
- \( 80(2.5) = 200 \)
Putting it all together: \[ h(2.5) = -100 + 200 + 30 = 130 \]
b) What does \( h(2.5) \) represent in the context of this problem?
In this context, \( h(2.5) = 130 \) represents the height of the rock, which is 130 inches above the ground, 2.5 seconds after it has been thrown.
c) Does it make sense to find \( h(-10) \) in the context of this problem?
Correct Answer: No, it does not make sense because time cannot be negative. Time \( t \) in this problem represents the duration after the rock has been thrown, and logically, you cannot have a negative time in this scenario.
Summary:
- \( h(2.5) = 130 \)
- It represents the height of the rock above the ground at 2.5 seconds.
- Finding \( h(-10) \) does not make sense because negative time is not applicable in this context.