Question

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.
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?

Possible answers
:: The time it takes for the height to be 2.5 inches above ground.
:: Yes, it makes sense because the height could be under sea level.
:: No, it does not make sense because we cannot have negative inches.
:: The height of the rock 2.5 seconds after it has been thrown.
No, it does not make sense because time cannot be negative.
The time it takes for the rock to hit the ground once it has been thrown.
2.5 seconds
190
30
130

Answer 1

To solve the problems, we will first evaluate \( h(2.5) \).

### a) Find \( h(2.5) \).

Using the given function:

\[
h(t) = -16t^2 + 80t + 30
\]

Substituting \( t = 2.5 \):

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

Calculating \( (2.5)^2 \):

\[
(2.5)^2 = 6.25
\]

So now we substitute:

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

Calculating \( -16(6.25) \):

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

Calculating \( 80(2.5) \):

\[
80 \times 2.5 = 200
\]

Putting it all together:

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

Thus,

\[
h(2.5) = 130
\]

### b) What does \( h(2.5) \) represent in the context of this problem?

In the context of this problem, \( h(2.5) \) represents **the height of the rock 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 because time cannot be negative.

### Summary of Answers:

- a) \( h(2.5) = 130 \)
- 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.