The Maximum Height of a Drone Taking Off

What is the maximum height of a drone modeled by the equation h(t) = -3t² + 12t + 96?

The maximum height of the drone is 168 meters.

Understanding the Drone Height Model

The equation provided reflects the height of the drone above the ground at different time intervals. In this case, the equation h(t) = -3t² + 12t + 96 is a quadratic equation where t represents time in minutes and h(t) represents the height in meters. The coefficient -3 of t² indicates that the height decreases as time progresses due to the negative sign in front of t².

Finding the Maximum Height

To determine the maximum height of the drone, we need to find the vertex of the quadratic equation. The vertex of a parabola is the highest or lowest point, depending on the sign of the coefficient of t². In this case, since the coefficient is negative, the vertex represents the maximum height.

Using the formula t = -b / (2a), where a = -3 and b = 12, we can calculate the time value at which the drone reaches its maximum height. Substituting the values, we get t = -12 / (2 * -3) = 2 minutes.

Once we have the time value, we can plug it back into the original equation to find the corresponding height. By substituting t = 2 into h(t) = -3t² + 12t + 96, we get h(2) = -3(2)² + 12(2) + 96 = 48 + 24 + 96 = 168 meters. Therefore, the maximum height of the drone taking off from the bridge is 168 meters.