Answer:
256 feet
Step-by-step explanation:
To find the maximum height, we have to find the vertex of the parabola, which is formed by the situation.
First, find the axis of symmetry using -(b/2a). This will give us the x value of the vertex.
Plug in b into the equation:
-(b/2a)
-(96 / 2(-16))
-(96 / -32)
-(-3)
= 3
Now, plug in 3 as x into the given equation, to find the maximum height that the rock will reach:
H(t) = -16t² + 96t + 112
H(3) = -16(3²) + 96(3) + 112
H(3) = 256
So, the maximum height that the rock will reach is 256 feet
Maximum height achieved by the rock on the parabolic path will be 256 ft.
Expression modeling the height of the rock is given by,
Since, this equation is a quadratic equation, path followed by the rock will be a parabolic path and the highest point attained by the rock will be the vertex of the parabola.
And the vertex will be → [tex][-\frac{b}{2a},H(-\frac{b}{2a})][/tex]
Compare the given equation with standard form of the quadratic equation.
[tex]y=ax^{2}+bx+c[/tex]
By comparing the equations → a = -16, b = 96 and c = 112
Therefore, x-coordinate → [tex]-\frac{b}{2a}=-\frac{96}{(-2\times 16)}[/tex]
[tex]=3[/tex]
And y-coordinate → [tex]H(-\frac{b}{2a})=H(3)[/tex]
[tex]=-16(3)^2+96(3)+112[/tex]
[tex]=-144+288+112[/tex]
[tex]=256[/tex]
Therefore, vertex of the parabola will be → (3, 256)
Here, x-coordinate represents the period or time taken by the rock to reach the highest point and y-coordinate represents the height achieved.
Therefore, maximum height the rock will reach will be 256 feet.
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