From the vertex, the quadratic equation is given by:
[tex]y = -1.5x^2 + 48x[/tex]
A quadratic equation has the following format:
[tex]y = ax^2 + bx + c[/tex]
- In this problem, it is concave-down, thus [tex]a < 0[/tex].
- The rocket is launched from the ground, thus [tex]c = 0[/tex].
Thus:
[tex]y = ax^2 + bx[/tex]
Now, we need to find the value b.
The x-value of the vertex is given by:
[tex]x_V = -\frac{b}{2a}[/tex]
In this problem, [tex]x_V = 16[/tex], thus:
[tex]b = -2ax_V = -2a(16) = -32a[/tex]
The y-value of the vertex is given by:
[tex]y_V = -\frac{\Delta}{4a} = -\frac{b^2 - 4ac}{4a}[/tex]
In this problem, [tex]y_V = 384[/tex], thus:
[tex]-\frac{b^2}{4a} = 384[/tex]
[tex]\frac{b^2}{4a} = -384[/tex]
Since [tex]b = -32a[/tex]:
[tex](-32a)^2 = -1536a[/tex]
[tex]1024a^2 + 1536a = 0[/tex]
[tex]a(1024a + 1536) = 0[/tex]
[tex]a \neq 0[/tex], thus:
[tex]1024a = -1536[/tex]
[tex]a = -\frac{1536}{1024}[/tex]
[tex]a = -1.5[/tex]
Then
[tex]b = -32a = -32(-1.5) = 48[/tex]
Thus, the equation is:
[tex]y = -1.5x^2 + 48x[/tex]
A similar problem is given at https://brainly.com/question/24626341
