A rock is thrown from one side of a river to another. The function H (T) = - 16t^2 + 80t + 30 gives the height in inches of the rock t seconds after it has been thrown.

Question

A. Find H (2.5) _
B. What does h (2.5) represent in the context of this problem? _
C. Does it make sense to find h (-10) in the context of this problem? _
Answer choices: 130, 30, 190, 2.5 seconds, the time it takes for the height to be 2.5 inches above ground, the height of the rock 2.5 seconds after it has been thrown, the time it takes for the rock to hit the ground once it has been thrown, Yes it makes sense because the height could be under sea level, no it does not make sense because time cannot be negative, no it does not make sense because we cannot have negative inches

Answer 1

Given the function \( H(t) = -16t^2 + 80t + 30 \), let's solve each part:

A. Find \( H(2.5) \)

To find \( H(2.5) \), substitute \( t = 2.5 \) into the function:

\[ H(2.5) = -16(2.5)^2 + 80(2.5) + 30 \]

Calculating \( (2.5)^2 \):

\[ (2.5)^2 = 6.25 \]

Now substituting this back into the equation:

\[ H(2.5) = -16(6.25) + 80(2.5) + 30 \]

Calculating \( -16(6.25) \):

\[ -16 \times 6.25 = -100 \]

Calculating \( 80(2.5) \):

\[ 80 \times 2.5 = 200 \]

Now substituting these values back into the function:

\[ H(2.5) = -100 + 200 + 30 \] \[ H(2.5) = 100 + 30 = 130 \]

So, \( H(2.5) = 130 \).

B. What does \( H(2.5) \) represent in the context of this problem?

\( H(2.5) \) represents the height of the rock 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?

No, it does not make sense to find \( H(-10) \) because time cannot be negative in this context — the negative time would imply a moment before the rock was thrown.

Summary of Answers:

A. 130
B. The height of the rock 2.5 seconds after it has been thrown
C. No, it does not make sense because time cannot be negative.