To find the maximum height reached by the rocket, we need to analyze the given quadratic equation:
\[ y = -16x^2 + 261x + 130. \]
Since the coefficient of the \( x^2 \) term is negative (-16), the parabola opens downward, indicating that the vertex of the parabola represents the maximum height of the rocket.
The general form of a quadratic equation is:
\[ y = ax^2 + bx + c, \]
where \( a \), \( b \), and \( c \) are constants. The vertex \( (x_v, y_v) \) of a parabola given by a quadratic equation can be found using the following formulas:
\[ x_v = -\frac{b}{2a}, \]
\[ y_v = ax_v^2 + bx_v + c. \]
Let's calculate the vertex of our quadratic equation.
1. First, we find the \( x \)-coordinate of the vertex, \( x_v \):
\[ x_v = -\frac{b}{2a} = -\frac{261}{2(-16)} = \frac{261}{32} = 8.15625 \text{ seconds}.\]
This value represents the time after launch when the rocket reaches its maximum height.
2. Next, we find the \( y \)-coordinate of the vertex, \( y_v \), which represents the maximum height:
\[ y_v = -16(x_v)^2 + 261(x_v) + 130. \]
Substituting \( x_v \) by 8.15625 seconds:
\[ y_v = -16(8.15625)^2 + 261(8.15625) + 130 = 1194.390625 \text{ feet}. \]
To express the maximum height to the nearest hundredth of a foot, we round the value of \( y_v \):
\[ y_{\text{max}} = 1194.39 \text{ feet}. \]
So, after approximately 8.16 seconds, the rocket reaches its maximum height of approximately 1194.39 feet. This is the highest point that the rocket will reach before it starts descending back towards the ground.
