Question

A rock is thrown from one side of a river to another. The function h(t)=−16t2+35t+80h(t)=−16t2+35t+80 gives the height of the rock t seconds after it has been thrown.(3 points) 1. Find h(2)h(2). 2. What does h(2)h(2) represent in the context of this problem? 3. Does it make sense to find h(−4)h(−4) in the context of this problem?

Answer 1

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.