Question

Paul stands on a roof of a building and throws a ball upwards into the air. The height, in feet, of the ball after t seconds is represented by the function h(t)=-16t^2+48t+64 . Paul wants to determine the height of the ball after 3 seconds and after 6 seconds.

Determine whether each statement below is true or false regarding height of the ball after 3 seconds and after 6 seconds. Select True or False for each statement.
Statement
True
False
The height of the ball after 3 seconds is 112 feet.
The height of the ball after 6 seconds is 224 feet.
The value of h(3) represents the height of the ball after 3 seconds.
The value of h(6) has no meaning as the height of the ball cannot be negative.

Answer 1

To evaluate the statements, we need to calculate the height of the ball using the function \( h(t) = -16t^2 + 48t + 64 \).

1. **Calculate \( h(3) \)**:
\[
h(3) = -16(3)^2 + 48(3) + 64
\]
\[
= -16(9) + 144 + 64
\]
\[
= -144 + 144 + 64
\]
\[
= 64 \text{ feet}
\]

2. **Calculate \( h(6) \)**:
\[
h(6) = -16(6)^2 + 48(6) + 64
\]
\[
= -16(36) + 288 + 64
\]
\[
= -576 + 288 + 64
\]
\[
= -576 + 352
\]
\[
= -224 \text{ feet}
\]

Now let's evaluate the statements:

1. **The height of the ball after 3 seconds is 112 feet.**
\( h(3) = 64 \) feet, not 112 feet.
**False**

2. **The height of the ball after 6 seconds is 224 feet.**
\( h(6) = -224 \) feet, which means it is below ground level (not physically meaningful in this context).
**False**

3. **The value of h(3) represents the height of the ball after 3 seconds.**
\( h(3) = 64 \) feet, which correctly represents the height after 3 seconds.
**True**

4. **The value of h(6) has no meaning as the height of the ball cannot be negative.**
While we calculated \( h(6) \) and found it to be -224 feet (which indeed has no physical meaning), the statement is essentially true if interpreted that negative height does not apply in this context.
**True**

Summarizing the answers:

- The height of the ball after 3 seconds is 112 feet: **False**
- The height of the ball after 6 seconds is 224 feet: **False**
- The value of h(3) represents the height of the ball after 3 seconds: **True**
- The value of h(6) has no meaning as the height of the ball cannot be negative: **True**