Final answer:
The rocket reaches a maximum height of 41.45 meters by finding the vertex of the given quadratic equation, which represents the peak of the rocket's flight.
Step-by-step explanation:
To determine the maximum height the rocket will reach, we need to find the vertex of the parabola described by the height equation h(t) = 15 + 23t – 5t2. Since the coefficient of the t2 term is negative, the parabola opens downward, indicating that the vertex represents the maximum height.
The vertex form for a quadratic equation ax2 + bx + c is h(t) = a(t – h)2 + k, where (h, k) is the vertex of the parabola. To find h, we use the formula h = -b/(2a).
Substituting a = -5 and b = 23 into the formula gives us h = -23/(2 * -5) = 2.3 seconds. Now we find the maximum height by substituting back into the original h(t):
h(2.3) = 15 + 23(2.3) – 5(2.3)2
h(2.3) = 15 + 52.9 – 26.45
h(2.3) = 67.9 – 26.45
h(2.3) = 41.45 meters
Therefore, the rocket reaches a maximum height of 41.45 meters.