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) = -16t² + 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 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.	☐	☑️

Explanation:

To determine the validity of each statement, we analyze the function \( h(t) = -16t^2 + 48t + 64 \) for \( t = 3 \) and \( t = 6 \).

1. The height of the ball after 3 seconds is 112 feet.

• Step 1: Substitute \( t = 3 \) into \( h(t) \):

$$ h(3) = -16(3)^2 + 48(3) + 64 $$

• Step 2: Calculate each term:

$$ -16(9) = -144, \quad 48(3) = 144 $$

• Step 3: Sum the terms:

$$ h(3) = -144 + 144 + 64 = 64 $$

The height at \( t = 3 \) is 64 feet, not 112. So the statement is False.

2. The height of the ball after 6 seconds is 224 feet.

• Step 1: Substitute \( t = 6 \) into \( h(t) \):

$$ h(6) = -16(6)^2 + 48(6) + 64 $$

• Step 2: Calculate each term:

$$ -16(36) = -576, \quad 48(6) = 288 $$

• Step 3: Sum the terms:

$$ h(6) = -576 + 288 + 64 = -224 $$

The height at \( t = 6 \) is -224 feet (which is physically impossible, but mathematically, the function gives this value). It is not 224 feet. So the statement is False.

3. The value of \( h(3) \) represents the height of the ball after 3 seconds.

• By definition, \( h(t) \) gives the height of the ball at time \( t \) seconds. Thus, \( h(3) \) is the height at \( t = 3 \) seconds. This statement is True.

4. The value of \( h(6) \) has no meaning as the height of the ball cannot be negative.

• Mathematically, \( h(6) \) is defined (it equals -224), but physically, a negative height is impossible (the ball would be below the starting point, which may not make sense in this context). However, the function still outputs a value for \( t = 6 \), so \( h(6) \) has a mathematical meaning (even if it is physically unrealistic). Thus, the statement is False.

Final Answers:

1. The height of the ball after 3 seconds is 112 feet: \(\boldsymbol{\text{False}}\)
2. The height of the ball after 6 seconds is 224 feet: \(\boldsymbol{\text{False}}\)
3. The value of \( h(3) \) represents the height of the ball after 3 seconds: \(\boldsymbol{\text{True}}\)
4. The value of \( h(6) \) has no meaning as the height of the ball cannot be negative: \(\boldsymbol{\text{False}}\)

Answer:

To determine the validity of each statement, we analyze the function \( h(t) = -16t^2 + 48t + 64 \) for \( t = 3 \) and \( t = 6 \).

1. The height of the ball after 3 seconds is 112 feet.

• Step 1: Substitute \( t = 3 \) into \( h(t) \):

$$ h(3) = -16(3)^2 + 48(3) + 64 $$

• Step 2: Calculate each term:

$$ -16(9) = -144, \quad 48(3) = 144 $$

• Step 3: Sum the terms:

$$ h(3) = -144 + 144 + 64 = 64 $$

The height at \( t = 3 \) is 64 feet, not 112. So the statement is False.

2. The height of the ball after 6 seconds is 224 feet.

• Step 1: Substitute \( t = 6 \) into \( h(t) \):

$$ h(6) = -16(6)^2 + 48(6) + 64 $$

• Step 2: Calculate each term:

$$ -16(36) = -576, \quad 48(6) = 288 $$

• Step 3: Sum the terms:

$$ h(6) = -576 + 288 + 64 = -224 $$

The height at \( t = 6 \) is -224 feet (which is physically impossible, but mathematically, the function gives this value). It is not 224 feet. So the statement is False.

3. The value of \( h(3) \) represents the height of the ball after 3 seconds.

• By definition, \( h(t) \) gives the height of the ball at time \( t \) seconds. Thus, \( h(3) \) is the height at \( t = 3 \) seconds. This statement is True.

4. The value of \( h(6) \) has no meaning as the height of the ball cannot be negative.

• Mathematically, \( h(6) \) is defined (it equals -224), but physically, a negative height is impossible (the ball would be below the starting point, which may not make sense in this context). However, the function still outputs a value for \( t = 6 \), so \( h(6) \) has a mathematical meaning (even if it is physically unrealistic). Thus, the statement is False.

Final Answers:

1. The height of the ball after 3 seconds is 112 feet: \(\boldsymbol{\text{False}}\)
2. The height of the ball after 6 seconds is 224 feet: \(\boldsymbol{\text{False}}\)
3. The value of \( h(3) \) represents the height of the ball after 3 seconds: \(\boldsymbol{\text{True}}\)
4. The value of \( h(6) \) has no meaning as the height of the ball cannot be negative: \(\boldsymbol{\text{False}}\)