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 -16t to the power of 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?

1 answer

To find the height of the rock at \( t = 1 \) second after it leaves Kenny's hand, he uses the given height model \( h(t) = -16t^2 + 32t + 7 \). This is a quadratic equation that describes the height of the rock over time.

Kenny substitutes \( t = 1 \) into the equation:

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

Calculating each term:

First, calculate \( -16(1)^2 \): \[ -16(1)^2 = -16(1) = -16 \]
Next, calculate \( 32(1) \): \[ 32(1) = 32 \]
Finally, since the initial height was 7 feet, we will add this term: \[ h(1) = -16 + 32 + 7 \]

Now combine these values:

First, add \( -16 + 32 \): \[ -16 + 32 = 16 \]
Now, add this result to 7: \[ 16 + 7 = 23 \]

Therefore, the height of the rock at \( t = 1 \) second is \( 23 \) feet.

In conclusion, Kenny correctly calculated the height of the rock after 1 second by substituting \( t = 1 \) into the height equation and performing the arithmetic operations correctly, leading to the result of 23 feet.