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 + 80t + 48. 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 \( h(t) = -16t^2 + 80t + 48 \) (assuming the function was miswritten and the correct leading coefficient for a projectile is -16, as the original text might have a typo).

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

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

\( h(3) = -16(3)^2 + 80(3) + 48 \)

Step 2: Calculate each term

\( -16(9) = -144 \), \( 80(3) = 240 \)

Step 3: Sum the terms

\( h(3) = -144 + 240 + 48 = 144 \)
Since \( 144
eq 112 \), the statement is False.

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

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

\( h(6) = -16(6)^2 + 80(6) + 48 \)

Step 2: Calculate each term

\( -16(36) = -576 \), \( 80(6) = 480 \)

Step 3: Sum the terms

\( h(6) = -576 + 480 + 48 = -48 \)
Since \( -48
eq 224 \) (and height cannot be negative here, but the value itself is not 224), the statement is False.

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

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

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

While \( h(6) = -48 \) is negative (implying the ball has hit the ground or gone below ground level), \( h(6) \) still has mathematical meaning (it represents the output of the function at \( t = 6 \), even if it is not physically meaningful for height). However, the statement claims it "has no meaning"—this is False (the function’s output exists, even if it is not a valid height).

Final Answers (Matching the Table):

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: \(\boldsymbol{\text{False}}\)

(Note: If the original function was \( h(t) = -18t^2 + 80t + 48 \) (as written), recalculating:

• \( h(3) = -18(9) + 240 + 48 = -162 + 240 + 48 = 126 \) (still not 112),
• \( h(6) = -18(36) + 480 + 48 = -648 + 480 + 48 = -120 \) (still not 224). The conclusions remain the same.)

Answer:

To determine the validity of each statement, we analyze the function \( h(t) = -16t^2 + 80t + 48 \) (assuming the function was miswritten and the correct leading coefficient for a projectile is -16, as the original text might have a typo).

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

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

\( h(3) = -16(3)^2 + 80(3) + 48 \)

Step 2: Calculate each term

\( -16(9) = -144 \), \( 80(3) = 240 \)

Step 3: Sum the terms

\( h(3) = -144 + 240 + 48 = 144 \)
Since \( 144
eq 112 \), the statement is False.

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

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

\( h(6) = -16(6)^2 + 80(6) + 48 \)

Step 2: Calculate each term

\( -16(36) = -576 \), \( 80(6) = 480 \)

Step 3: Sum the terms

\( h(6) = -576 + 480 + 48 = -48 \)
Since \( -48
eq 224 \) (and height cannot be negative here, but the value itself is not 224), the statement is False.

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

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

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

While \( h(6) = -48 \) is negative (implying the ball has hit the ground or gone below ground level), \( h(6) \) still has mathematical meaning (it represents the output of the function at \( t = 6 \), even if it is not physically meaningful for height). However, the statement claims it "has no meaning"—this is False (the function’s output exists, even if it is not a valid height).

Final Answers (Matching the Table):

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: \(\boldsymbol{\text{False}}\)

(Note: If the original function was \( h(t) = -18t^2 + 80t + 48 \) (as written), recalculating:

• \( h(3) = -18(9) + 240 + 48 = -162 + 240 + 48 = 126 \) (still not 112),
• \( h(6) = -18(36) + 480 + 48 = -648 + 480 + 48 = -120 \) (still not 224). The conclusions remain the same.)