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(t) = -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 function that describes the height of the object: $s(t)=-16t^2+64t+100$.
2. Recognize that this is a quadratic function in the form $a{t}^2+bt+c$, where $a=-16$, $b=64$, and $c=100$.
3. To find the time $t$ at which the object reaches its maximum height, use the vertex formula for a parabola, $t=-\frac{b}{2a}$.
4. Substitute the values of $a$ and $b$ into the vertex formula: $t=-\frac{64}{2(-16)}$.
5. Calculate the maximum height by substituting the value of $t$ found in the previous step back into the original function $s(t)$.

Key Concepts

Quadratic Functions

The height of the object is modeled by a quadratic function, which is a polynomial of degree two. Quadratic functions have a parabolic shape and can be expressed in the standard form s(t) = at^2 + bt + c, where 'a', 'b', and 'c' are constants. The vertex of the parabola represents the maximum or minimum point, which is crucial for determining the maximum height of the object.

Vertex of a Parabola

The vertex of a parabola given by a quadratic function is the point where the function reaches its maximum or minimum value. For a downward-opening parabola (where 'a' is negative), the vertex can be found using the formula t = -b/(2a). This value of 't' gives the time at which the object reaches its maximum height, which is essential for solving the problem.

Maximum Height Calculation

To find the maximum height of the object, we first determine the time at which it occurs using the vertex formula. Once we have this time, we substitute it back into the original height function s(t) to calculate the maximum height. This process involves evaluating the quadratic function at the vertex, providing the highest point the object reaches during its flight.