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A volleyball is served from a height of 6 feet with an initial velocity of 33 feet per second. The situation is modeled using the function h=−6t2+33t+6

, where h is the height in feet and t is time in seconds. Which of the following uses the function to correctly identify the corresponding y-values to the given x-values of 0, 1, 2, 3, and 4?(1 point)
Responses

(0,0)

, (1,27), (2,42), (3,45), and (4,36)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma 27 right parenthesis , left parenthesis 2 comma 42 right parenthesis , left parenthesis 3 comma 45 right parenthesis , and left parenthesis 4 comma 36 right parenthesis
(0,6)
, (1,33), (2,48), (3,51) and (4,42)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 33 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 51 right parenthesis and left parenthesis 4 comma 42 right parenthesis
(0,6)
, (1,21), (2,48), (3,69), and (4,90)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 21 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 69 right parenthesis , and left parenthesis 4 comma 90 right parenthesis
(6,0)
, (33,1), (48,2), (51,3), and (42,4)

1 answer

We need to evaluate the given function \( h(t) = -6t^2 + 33t + 6 \) at the values \( t = 0, 1, 2, 3, \) and \( 4 \) to find the corresponding heights.

  1. For \( t = 0 \): \[ h(0) = -6(0)^2 + 33(0) + 6 = 6 \]

  2. For \( t = 1 \): \[ h(1) = -6(1)^2 + 33(1) + 6 = -6 + 33 + 6 = 33 \]

  3. For \( t = 2 \): \[ h(2) = -6(2)^2 + 33(2) + 6 = -6(4) + 66 + 6 = -24 + 66 + 6 = 48 \]

  4. For \( t = 3 \): \[ h(3) = -6(3)^2 + 33(3) + 6 = -6(9) + 99 + 6 = -54 + 99 + 6 = 51 \]

  5. For \( t = 4 \): \[ h(4) = -6(4)^2 + 33(4) + 6 = -6(16) + 132 + 6 = -96 + 132 + 6 = 42 \]

So the pairs of \( (t, h(t)) \) are:

  • \( (0, 6) \)
  • \( (1, 33) \)
  • \( (2, 48) \)
  • \( (3, 51) \)
  • \( (4, 42) \)

Now let's match these with the options given:

  • Option 1: \( (0,0), (1,27), (2,42), (3,45), (4,36) \) – Incorrect
  • Option 2: \( (0,6), (1,33), (2,48), (3,51), (4,42) \) – Correct
  • Option 3: \( (0,6), (1,21), (2,48), (3,69), (4,90) \) – Incorrect
  • Option 4: \( (6,0), (33,1), (48,2), (51,3), (42,4) \) – Incorrect

The correct response is:

(0, 6), (1, 33), (2, 48), (3, 51), and (4, 42).