Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by $s\left(t\right)=-16t^2+64t+100$. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

Verified step by step guidance

1. Identify the given quadratic function for height: \(s(t) = -16t^{2} + 64t + 100\). This represents the height of the object at time \(t\) seconds.
2. Recall that the graph of \(s(t)\) is a parabola opening downward (since the coefficient of \(t^{2}\) is negative), so the maximum height occurs at the vertex of the parabola.
3. Use the vertex formula for a quadratic function \(at^{2} + bt + c\), where the time \(t\) at the vertex is given by \(t = -\frac{b}{2a}\). Here, \(a = -16\) and \(b = 64\).
4. Calculate the time \(t\) to reach maximum height by substituting \(a\) and \(b\) into the vertex formula: \(t = -\frac{64}{2 \times (-16)}\).
5. Find the maximum height by substituting this value of \(t\) back into the original height function \(s(t)\): \(s(t) = -16t^{2} + 64t + 100\).

Key Concepts

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(t) = at^2 + bt + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. In this problem, the height function is quadratic, representing the object's height over time.

Vertex of a Parabola

The vertex of a parabola given by f(t) = at^2 + bt + c is the point where the function reaches its maximum or minimum value. For a downward-opening parabola (a < 0), the vertex represents the maximum point. The time to reach maximum height is found using t = -b/(2a).

Evaluating the Function at the Vertex

Once the time at which the maximum height occurs is found, substitute this value back into the height function s(t) to find the maximum height. This step gives the highest point the object reaches during its motion.