Question

Kenny is standing on top of a boulder at his neighborhood pond. He picks up a small rock, so that it is 7 feet off the ground, and throws the rock into the pond. The height of the rock after t seconds can be modeled by the expression -16^2 + 32t + 7. Kenny wants to know the height of the rock at 1 second after it leaves his hand, t = 1, and comes up with the answer 23 feet. How did Kenny find this result?

Answer 1

To find the height of the rock at \( t = 1 \) second, Kenny needs to substitute \( t = 1 \) into the given height equation:

\[
h(t) = -16t^2 + 32t + 7
\]

Now, plug in \( t = 1 \):

\[
h(1) = -16(1)^2 + 32(1) + 7
\]

Calculating each term:

1. Calculate \( -16(1)^2 \):
\[
-16(1) = -16
\]

2. Calculate \( 32(1) \):
\[
32(1) = 32
\]

3. The constant term is \( 7 \).

Now combine all these values:

\[
h(1) = -16 + 32 + 7
\]

Next, simplify this step-by-step:

1. Combine \( -16 \) and \( 32 \):
\[
-16 + 32 = 16
\]

2. Now add \( 7 \) to \( 16 \):
\[
16 + 7 = 23
\]

So, the height of the rock after \( 1 \) second is:

\[
h(1) = 23 \text{ feet}
\]

This confirms that Kenny correctly found the height of the rock to be 23 feet at \( t = 1 \) second.