To determine whether Jessica's rocket will clear 50 feet, we need to find the maximum height reached by the rocket. This maximum height can be determined by finding the vertex of the quadratic equation [tex]\(y = -3x^2 + 6x + 48\)[/tex].
The general form of a quadratic equation is [tex]\(y = ax^2 + bx + c\)[/tex], and the vertex form of this equation gives us the maximum (or minimum) point of the parabola. The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Where:
- [tex]\(a = -3\)[/tex]
- [tex]\(b = 6\)[/tex]
- [tex]\(c = 48\)[/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the vertex formula, we get:
[tex]\[ x = -\frac{6}{2(-3)} = -\frac{6}{-6} = 1 \][/tex]
So, the x-coordinate of the vertex is [tex]\(x = 1\)[/tex].
Next, we need to find the y-coordinate of the vertex by substituting [tex]\(x = 1\)[/tex] back into the original equation:
[tex]\[ y = -3(1)^2 + 6(1) + 48 \][/tex]
Performing the calculations step-by-step:
[tex]\[ y = -3(1) + 6(1) + 48 \][/tex]
[tex]\[ y = -3 + 6 + 48 \][/tex]
[tex]\[ y = 3 + 48 \][/tex]
[tex]\[ y = 51 \][/tex]
The y-coordinate of the vertex is [tex]\(y = 51\)[/tex]. Therefore, the maximum height of the rocket is 51 feet.
Since 51 feet is greater than 50 feet, we can conclude that Jessica's rocket will indeed clear 50 feet.
So, Jessica's rocket will clear 50 feet.
