Question

if a ball is thrown upward at 49 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 + 49t - 4.9t^{2} ), where ( t ) is the number of seconds after the ball is thrown. answer parts a and b
the ( t )-coordinate of the vertex of the graph of this quadratic function is ( t = 5 )
(simplify your answer.)
the ( s )-coordinate of the vertex of the graph of this quadratic function is 147.5
(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 height of 5 meters in 147.5 seconds.
b. the ball reaches its maximum height of 147.5 meters in 5 seconds.
c. the ball hits the ground after 6 seconds at a speed of 142.3 meters per second.
d. the ball reaches its maximum speed of 142.3 meters per second in 6 seconds.

Explanation:

For a quadratic function \(S = at^{2}+bt + c\) (in this case \(S=-4.9t^{2}+49t + 25\), so \(a=-4.9\), \(b = 49\), \(c = 25\)), the \(t\) - coordinate of the vertex is given by \(t=-\frac{b}{2a}\). Substituting \(a=-4.9\) and \(b = 49\) gives \(t = 5\). Then substituting \(t = 5\) into the function \(S=25+49t-4.9t^{2}\) gives \(S=25+49\times5-4.9\times5^{2}=25 + 245-122.5=147.5\). The vertex of a parabola \(y = ax^{2}+bx + c\) (here \(S\) is a function of \(t\)) gives the maximum (since \(a=-4.9<0\)) value of the function. The \(t\) - coordinate is the time and the \(S\) - coordinate is the height.

Answer:

B. The ball reaches its maximum height of 147.5 meters in 5 seconds.