Application of Nonlinear Inequalities

In a vertical projectile motion without air resistance, an object is launched from ground level with a velocity of 50 meters per second. Its height over time follows the quadratic equation: negative five t squared plus fifty t, where negative five is half the acceleration due to gravity. The goal is to determine the time interval when the projectile’s height exceeds 100 meters.

Since the height function is quadratic, comparing it to a constant value involves solving a quadratic inequality.

The inequality is rearranged by moving all terms to one side and dividing by negative five, which simplifies the leading coefficient to one.

Since the resulting expression cannot be factored easily, the quadratic formula is used. The coefficients are substituted into the formula to find two solution points.

These values represent the beginning and end of the time period during which the projectile stays above 100 meters.

A graph of the height function helps visualize this situation. The curve rises above the 100-meter mark, reaches a peak, and then falls back to the mark.

On the graph, the part of the curve that lies above 100 meters is shaded to show the time interval during which the condition is met.