Question

if a ball is thrown upward at 39.2 meters per second from the top of a building that is 25 meters high, the height of the ball can be modeled by ( s = 25 + 39.2t - 4.9t^{2} ), where ( t ) is the number of seconds after the ball is thrown.
answer parts a through c.
a. find the t - coordinate and s - coordinate of the vertex of the graph of this quadratic function.
the t - coordinate of the vertex of the graph of this quadratic function is ( t = 4 )
(simplify your answer)
the s - coordinate of the vertex of the graph of this quadratic function is ( 103.4 )
(simplify your answer)
b. explain the meaning of the coordinates of the vertex for this model. choose the correct explanation below.
○ a. the ball reaches its maximum speed of 98.2 meters per second in 5 seconds
○ b. the ball reaches its maximum height of 103.4 meters in 4 seconds.
○ c. the ball hits the ground after 5 seconds at a speed of 98.2 meters per second.
○ d. the ball reaches its maximum height of 4 meters in 103.4 seconds.

Explanation:

Step1: Analyze the quadratic function form

The quadratic function is \(S = - 4.9t^{2}+39.2t + 25\), where \(a=-4.9\), \(b = 39.2\), \(c = 25\). For a quadratic function \(y=ax^{2}+bx + c\), the \(t\)-coordinate of the vertex is \(t=-\frac{b}{2a}\).

Step2: Calculate the \(t\)-coordinate

Substitute \(a=-4.9\) and \(b = 39.2\) into \(t =-\frac{b}{2a}\), we get \(t=-\frac{39.2}{2\times(-4.9)}=\frac{39.2}{9.8}=4\).

Step3: Calculate the \(S\)-coordinate

Substitute \(t = 4\) into \(S=-4.9t^{2}+39.2t + 25\), \(S=-4.9\times4^{2}+39.2\times4 + 25=-4.9\times16+156.8+25=-78.4+156.8+25 = 103.4\).

Step4: Interpret the vertex coordinates

Since the coefficient of \(t^{2}\) (\(a=-4.9<0\)) in the quadratic function \(S(t)\), the parabola opens down - ward. The vertex \((t,S)=(4,103.4)\) represents the maximum point of the function. Here \(t\) is the time and \(S\) is the height.

Answer:

B. The ball reaches its maximum height of 103.4 meters in 4 seconds.