Final answer:
To determine the splashdown time of the rocket, the quadratic formula is applied to the height function h(t), resulting in a positive time value when rounded to two decimals. To ascertain the rocket's peak height, we calculate the vertex of the parabola represented by h(t) and evaluate h(t) at that time, once again rounding to two decimals.
Step-by-step explanation:
To find the time of splashdown, when the rocket hits the ocean, we need to solve the equation h(t) = 0, where h(t) = -4.9t² + 94t + 129, for time t. To do this, we will apply the quadratic formula, which is t = (-b ± sqrt(b² - 4ac)) / (2a) for the quadratic equation at² + bt + c = 0. In our equation, a = -4.9, b = 94, and c = 129.
Plugging these values into the quadratic formula gives us two possible solutions for t, of which the positive value will be the time of splashdown:
t = (-94 ± sqrt(94² - 4(-4.9)(129))) / (2(-4.9))
We take the positive square root because the launch occurs at t = 0 and we are interested in the future event of splashdown. After calculating, we round our answer to two decimals.
To find out how high above sea-level the rocket gets at its peak, we need to determine the vertex of the parabola described by the equation h(t). Since the coefficient of t² is negative, the parabola opens downward and its vertex is the peak. The formula for the time at which the vertex occurs is t = -b/(2a) for the quadratic equation at² + bt + c.
The peak height will then be h(t) at this vertex time. We evaluate h(t) using this t value to find the maximum height, rounding our answer to two decimals.