Fireworks are launched into the air. The quadratic function y=-16x^2+197x+6 models the fireworks" height, y, in feet, x seconds after they are launched. When should the fi explode so that they go off at the greatest height? What is that height? (Round answers the nearest hundredth.) A. 1092.00sec,6.00ft B. 6.00sec,197.00ft C. -6.16sec,600.39ft D. 6.16sec,612.39ft

To find when the fireworks should explode to reach the greatest height, we need to determine the vertex of the quadratic function. The vertex of a parabola in the form $y = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$.<br /><br />In this case, the quadratic function is $y = -16x^2 + 197x + 6$. Plugging in the values of $a$ and $b$, we get:<br /><br />$x = -\frac{197}{2(-16)} = \frac{197}{32} \approx 6.16$<br /><br />So, the fireworks should explode approximately 6.16 seconds after they are launched.<br /><br />To find the height at which the fireworks explode, we substitute the value of $x$ back into the original equation:<br /><br />$y = -16(6.16)^2 + 197(6.16) + 6$<br /><br />$y \approx 600.39$<br /><br />Therefore, the fireworks should explode approximately 6.16 seconds after they are launched, and the height at which they explode is approximately 600.39 feet.<br /><br />The correct answer is D. $6.16sec,600.39ft$.