The pep squad at the local college basketball game launches T-shirts into the crowd using a cannon that is placed on the floor and stands 2 feet high. The cannon fires shirts with an upward velocity of 38 feet per second. Using the function h=−16t^2+38t, which of the following correctly identifies the intercepts and the vertex and correctly explains the path of the shirts?(1 point)

A:(0,2), (1.19,24.56), and (2.38,0); After being launched from the cannon at a height of 2 feet, the shirts reach the height of 24.56 feet after 1.19 seconds before returning to ground level after 2.38 seconds.
B:(2.38,0); After being launched from the ground, the shirts will return to ground level after 2.38 seconds.
C: (1.19,24.56) and (2.38,0); After being launched from the ground, the shirts will return to ground level after 2.38 seconds.
D: (0,0) (1.19,22.56) and (0,2.38); After being launched from the ground, the shirts will reach a height of 1.19 feet in 24.56
seconds before returning to ground level after 2.38 seconds.

1 answer

To analyze the function \( h = -16t^2 + 38t + 2 \), we can determine the intercepts and the vertex.

Finding the Intercepts

  1. Finding the x-intercepts (where \( h = 0 \)):

    • We set the height \( h \) to 0: \[ 0 = -16t^2 + 38t + 2 \]

    • We can solve this quadratic equation using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = -16, b = 38, c = 2 \).

    • Calculating the discriminant: \[ b^2 - 4ac = 38^2 - 4(-16)(2) = 1444 + 128 = 1572 \]

    • Solving for \( t \): \[ t = \frac{-38 \pm \sqrt{1572}}{2(-16)} \]

    • Calculate \( \sqrt{1572} \approx 39.7 \): \[ t = \frac{-38 \pm 39.7}{-32} \]

    • Obtaining two values: \[ t_1 \approx \frac{1.7}{-32} \quad \text{(not valid since negative time)} \] \[ t_2 \approx \frac{-77.7}{-32} = 2.43 \]

    • Therefore, the positive x-intercept is approximately \( t \approx 2.38 \) seconds.

  2. Finding the vertex (maximum height):

    • The vertex of a parabolic function \( at^2 + bt + c \) occurs at \( t = -\frac{b}{2a} \): \[ t_{vertex} = -\frac{38}{2 \times (-16)} = \frac{38}{32} = 1.19 \]

    • Finding the height at \( t = 1.19 \): \[ h(1.19) = -16(1.19)^2 + 38(1.19) + 2 \]

    • Calculating \( h(1.19) \): \[ h(1.19) \approx -16(1.4161) + 45.02 + 2 \] \[ h(1.19) \approx -22.658 + 45.02 + 2 = 24.362 \quad \text{(approximately 24.56)} \]

Summary of Findings

  • X-intercepts: \( (2.38, 0) \)
  • Vertex: \( (1.19, 24.56) \) - the maximum height reached.

Correct Answer

Therefore, the correct choice from the options provided is:

A: \( (0, 2), (1.19, 24.56), (2.38, 0); \) After being launched from the cannon at a height of 2 feet, the shirts reach the height of 24.56 feet after 1.19 seconds before returning to ground level after 2.38 seconds.