A person who is standing on a ledge throws a rock into the air. The rock reaches a maximum height of 676 feet

above the ground after 1.5 seconds.

If the rock hits the ground below the ledge 8 seconds after it is thrown, which quadratic function can be used to find

the height of the rock above the ground t seconds after it is thrown?

Answer:

The quadratic equation is [tex]h(t) = -318.22t^2 + 954.67t + 12728.89[/tex]

Step-by-step explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

[tex]f(x) = ax^{2} + bx + c[/tex]

It's vertex is the point [tex](x_{v}, y_{v})[/tex]

In which

[tex]x_{v} = -\frac{b}{2a}[/tex]

[tex]y_{v} = -\frac{\Delta}{4a}[/tex]

Where

[tex]\Delta = b^2-4ac[/tex]

If a<0, the vertex is a maximum point, that is, the maximum value happens at [tex]x_{v}[/tex], and it's value is [tex]y_{v}[/tex].

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

[tex]ax^{2} + bx + c, a\neq0[/tex].

This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:

[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]

[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]

[tex]\Delta= b^{2} - 4ac[/tex]

Quadratic equation:

[tex]ax^2 + bx + c = 0[/tex]

The rock reaches a maximum height of 676 feet above the ground after 1.5 seconds.

This means that:

[tex]-\frac{b}{2a} = 1.5[/tex]

[tex]y_{v} = -\frac{b^2-4ac}{4a} = 676[/tex]

If the rock hits the ground below the ledge 8 seconds after it is thrown

This means that:

[tex]64a + 8b + c = 0[/tex]

Relation of b and a:

[tex]-\frac{b}{2a} = 1.5[/tex]

This means that:

[tex]b = -3a[/tex]

Relationship of c and a:

[tex]64a + 8b + c = 0[/tex]

[tex]64a - 24a + c = 0[/tex]

[tex]c = -40a[/tex]

Finding a:

[tex]y_{v} = -\frac{(-3a)^2-4a(-40a)}{4a} = 676[/tex]

[tex]9a^2+160a = -2704a[/tex]

[tex]9a^2 + 2864a = 0[/tex]

[tex]a(9a + 2864) = 0[/tex]

Since it is a quadratic equation, a cannot be 0.

[tex]9a + 2864 = 0[/tex]

[tex]9a = -2864[/tex]

[tex]a = -\frac{2864}{9}[/tex]

[tex]a = -318.22[/tex]

Finding b and c:

[tex]b = -3a = -3(-318.22) = 954.67[/tex]

[tex]c = -40a = -40(-318.22) = 12728.89[/tex]

The quadratic equation is given by:

[tex]h(t) = -318.22t^2 + 954.67t + 12728.89[/tex]

The quadratic equation is [tex]h(t) = -318.22t^2 + 954.67t + 12728.89[/tex]

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