Question

At baseball practice, Matt hit the ball into the outfield. The function h(t)=−16t2+32t+3
gives the height h(t)
of the baseball in feet, t
seconds after it is hit.

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

B) What does h(1.74)
represent in context? Response area

C) Which value would not be valid in the context of this problem?

Answer 1

To solve the problem, let's analyze the function \( h(t) = -16t^2 + 32t + 3 \) and respond to each query accordingly.

### A) What is the value of h(1)?

To find \( h(1) \), we substitute \( t = 1 \) into the function:

\[
h(1) = -16(1)^2 + 32(1) + 3
\]
\[
= -16(1) + 32 + 3
\]
\[
= -16 + 32 + 3
\]
\[
= 16 + 3
\]
\[
= 19
\]

So the value of \( h(1) \) is **19 feet**.

### B) What does h(1.74) represent in context?

To find \( h(1.74) \), we can calculate it, but first, we determine what it represents:

\( h(1.74) \) represents the height of the baseball above the ground at \( t = 1.74 \) seconds after it has been hit.

Now, let's calculate \( h(1.74) \):

\[
h(1.74) = -16(1.74)^2 + 32(1.74) + 3
\]
Calculating each part:
\[
= -16(3.0276) + 32(1.74) + 3
\]
\[
= -48.4416 + 55.68 + 3
\]
\[
= -48.4416 + 58.68
\]
\[
= 10.2384 \text{ feet}
\]

Thus, \( h(1.74) \approx 10.24 \) feet.

In summary, \( h(1.74) \) represents the height of the baseball at approximately **10.24 feet** 1.74 seconds after being hit.

### C) Which value would not be valid in the context of this problem?

In the context of the problem, \( t \) represents time in seconds after the baseball is hit. Therefore, valid values for \( t \) must be non-negative (i.e., \( t \geq 0 \)) because time cannot be negative.

Any negative time value (e.g., \( t = -1, -2, \) etc.) would not be valid in this context as it does not correspond to a realistic scenario of any event occurring before the ball is hit.

Thus, any negative value of \( t \) would not be valid. An example might be \( t = -1 \).