A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot.
y, equals, minus, 16, x, squared, plus, 198, x, plus, 77
y=−16x
2
+198x+77
To find the maximum height reached by the rocket, we need to determine the vertex of the parabolic equation y = -16x^2 + 198x + 77.
The x-coordinate of the vertex of a parabolic equation in the form y = ax^2 + bx + c is given by x = -b/(2a). In this case, a = -16 and b = 198.
x = -198 / (2*(-16))
x = -198 / (-32)
x = 6.1875 (rounded to four decimal places)
Now, substitute x = 6.1875 back into the original equation to find the corresponding y-coordinate:
y = -16(6.1875)^2 + 198(6.1875) + 77
y = -16(38.4277) + 1229.8125 + 77
y = -614.8432 + 1229.8125 + 77
y = 692.9693
Therefore, the maximum height reached by the rocket is approximately 692.97 feet.