Questions: The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The T-shirts have an upward velocity of 30 feet per second. Using the function y=-16 t^2+30 t+2, which of the following correctly identifies the vertex of the parabola and best explains its meaning? (1 point) The vertex is at (16.063,0.9375). The shirts will reach a maximum height of roughly 0.928 feet. The vertex is at (0.9375,16.063). The shirts will reach a maximum height of roughly 0.928 feet. The vertex is at (16.063,0.938). The shirts will reach a maximum height of roughly 16 feet. The vertex is at (0.9375,16.063). The shirts will reach a maximum height of roughly 16 feet.

The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The T-shirts have an upward velocity of 30 feet per second. Using the function y=-16 t^2+30 t+2, which of the following correctly identifies the vertex of the parabola and best explains its meaning? (1 point)
The vertex is at (16.063,0.9375). The shirts will reach a maximum height of roughly 0.928 feet.
The vertex is at (0.9375,16.063). The shirts will reach a maximum height of roughly 0.928 feet.
The vertex is at (16.063,0.938). The shirts will reach a maximum height of roughly 16 feet.
The vertex is at (0.9375,16.063). The shirts will reach a maximum height of roughly 16 feet.

Transcript text: The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The $T$-shirts have an upward velocity of 30 feet per second. Using the function $y=-16 t^{2}+30 t+2$, which of the following correctly identifies the vertex of the parabola and best explains its meaning? (1 point) The vertex is at $(16.063,0.9375)$. The shirts will reach a maximum height of roughly 0.928 feet. The vertex is at $(0.9375,16.063)$. The shirts will reach a maximum height of roughly 0.928 feet. The vertex is at $(16.063,0.938)$. The shirts will reach a maximum height of roughly 16 feet. The vertex is at $(0.9375,16.063)$. The shirts will reach a maximum height of roughly 16 feet.

Solution

Solution Steps

Step 1: Identify the Vertex of the Parabola

The given function is $ y = -16t^2 + 30t + 2 $. This is a quadratic function in the form $ y = at^2 + bt + c $, where $ a = -16 $, $ b = 30 $, and $ c = 2 $.

The vertex of a parabola given by $ y = ax^2 + bx + c $ can be found using the formula for the time at which the vertex occurs:

\[ t = -\frac{b}{2a} \]

Substituting the values of $ a $ and $ b $:

\[ t = -\frac{30}{2 \times (-16)} = \frac{30}{32} = 0.9375 \]

Step 2: Calculate the Maximum Height

Substitute $ t = 0.9375 $ back into the original equation to find the maximum height:

\[ y = -16(0.9375)^2 + 30(0.9375) + 2 \]

Calculate $ (0.9375)^2 $:

\[ (0.9375)^2 = 0.87890625 \]

Substitute back into the equation:

\[ y = -16(0.87890625) + 30(0.9375) + 2 \]

Calculate each term: