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)

Question

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.

GPT-4o mini · 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.