Question

If a baseball is projected upward from ground level with an initial velocity of 32 feet per​ second, then its height is a function of ​time, given by s=−16t²+32t. What is the maximum height reached by the​ ball?

Answer 1

To solve this problem, we first need to understand that the function s = -16t² + 32t describes a parabola. The maximum point in a parabola is always located at the vertex - formally known in mathematics as the maximum or minimum of the function.

The vertex of a parabola defined by the equation y = ax² + bx + c occurs at the point h = -b / (2a). In this case, a is the coefficient of t² and b is the coefficient of t, so a = -16 and b = 32.

Substituting these values into the equation, we get h = -32 / (2 * -16), which simplifies to h = 1. This value tells us at what time the baseball reaches its maximum height.

To find the maximum height, we substitute h = 1 (the time when the baseball reaches its maximum height) back into the original equation. This gives us s = -16(1)² + 32(1), simplifying to s = -16 + 32 = 16 feet.

Therefore, the maximum height reached by the baseball is 16 feet.