Question

a ball is thrown straight upward. a position function ( s(t)=\frac{1}{2} g t^{2}+v_{0} t+s_{0} ) can be used to describe the balls height, ( s(t) ), in feet, after ( t ) seconds. a graph of the balls height is shown to the right. complete parts a through c below. ( s(3.25)= ) (simplify your answer.) interpret ( s(3.25) ). choose the correct answer below. a. the ball hits the ground 3.25 seconds after it is thrown. b. the ball is below ground 3.25 seconds after it is thrown. c. the ball is half its highest height 3.25 seconds after it is thrown. d. the ball hits its highest height 3.25 seconds after it is thrown. c. after how many seconds does the ball reach its maximum height? what is its maximum height? the ball reaches its maximum height after seconds (type an integer or a decimal)

Explanation:

Step1: Analyze the graph

The graph of the position - function \(s(t)\) is a parabola. For a parabola \(y = ax^{2}+bx + c\) (in our case \(s(t)=\frac{1}{2}gt^{2}+v_{0}t + s_{0}\), which is a quadratic function of the form \(y = at^{2}+bt + c\) with \(a=\frac{1}{2}g\), \(b = v_{0}\), \(c=s_{0}\)), the vertex of the parabola gives the maximum height of the ball.

Step2: Locate the vertex

The vertex of a parabola \(y = ax^{2}+bx + c\) has its \(x\) - coordinate (in our case \(t\) - coordinate) given by \(t=-\frac{b}{2a}\). But from the graph, we can directly read the coordinates of the vertex. The vertex of the graph \(s(t)\) (the point where the ball reaches its maximum height) has the \(t\) - value (time) and \(s(t)\) - value (height). Looking at the graph, the vertex is at the point \((2,40)\)

Answer:

The ball reaches its maximum height after \(2\) seconds.