A rocket follows a parabolic trajectory. After t seconds, the vertical height of the rocket above the ground, in meters, is given

H(t)= 37t-t^2

well, you know it hit the ground after 37 seconds, right?

The max height was reached at t = 37/2 seconds.

To find the maximum height reached by the rocket, we need to determine the vertex of the parabolic trajectory. The equation for the height of the rocket at any given time is given by H(t) = 37t - t^2.

The vertex of a parabola in the form y=ax^2+bx+c can be found by using the formula x = -b/2a.

In the case of the height function H(t) = 37t - t^2, we can see that a = -1, b = 37. Plugging these values into the formula, we get:

t = -(37) / (2*(-1))
t = 37 / 2

Therefore, the time at which the rocket reaches its maximum height is t = 37 / 2 seconds.

To find the maximum height, we substitute this value back into the height function:

H(t) = 37t - t^2
H(37 / 2) = 37 * (37 / 2) - (37 / 2)^2

Simplifying this expression, we get:

H(37 / 2) = (37 * 37) / 2 - (37^2) / 4

H(37 / 2) = 1369 / 2 - 1369 / 4
H(37 / 2) = (1369 * 2 - 1369) / 4
H(37 / 2) = 2738 - 1369 / 4
H(37 / 2) = 1369 / 4
H(37 / 2) = 342.25

Therefore, the maximum height reached by the rocket is 342.25 meters.