Question

A basketball is shot from the free-throw line. The function h(t)=−16t2+40t+6
gives the height h(t)
of the basketball in feet, t
seconds after it is released.

What is the value of h(1)
? Response area

What does h(2.5)
represent in the context of this problem? Response area

What does h(−1)
represent in the context of this problem?

Answer 1

To solve the problems, we will evaluate the function \( h(t) = -16t^2 + 40t + 6 \).

### 1. What is the value of \( h(1) \)?

We substitute \( t = 1 \) into the function:

\[
h(1) = -16(1)^2 + 40(1) + 6
\]
\[
= -16(1) + 40 + 6
\]
\[
= -16 + 40 + 6
\]
\[
= 24
\]

So, \( h(1) = 24 \) feet.

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

To find out what \( h(2.5) \) represents, we calculate \( h(2.5) \):

\[
h(2.5) = -16(2.5)^2 + 40(2.5) + 6
\]
\[
= -16(6.25) + 100 + 6
\]
\[
= -100 + 100 + 6
\]
\[
= 6
\]

Thus, \( h(2.5) = 6 \) feet.

In the context of the problem, \( h(2.5) \) represents the height of the basketball 2.5 seconds after it is released. The height is 6 feet at that time.

### 3. What does \( h(-1) \) represent in the context of this problem?

To interpret \( h(-1) \), we could calculate it as follows:

\[
h(-1) = -16(-1)^2 + 40(-1) + 6
\]
\[
= -16(1) - 40 + 6
\]
\[
= -16 - 40 + 6
\]
\[
= -50
\]

So, \( h(-1) = -50 \) feet.

In the context of this problem, \( h(-1) \) would represent the height of the basketball at a time that is 1 second before it is released. Since time cannot physically be negative in this scenario, it indicates a hypothetical situation before the ball was released and the negative height values don't have a practical meaning in this context; they suggest that the ball would be below ground level (which isn't possible in a practical setting). Thus, it indicates a point in time before the shot was taken, but it doesn't reflect any physical reality related to the ball's actual trajectory.