Question

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 1

The quadratic equation is
`h(t) = -318.22t^2 + 954.67t + 12728.89`

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), y_(v))`

In which

`x_(v) = -(b)/(2a)`

`y_(v) = -(\Delta)/(4a)`

Where

`\Delta = b^2-4ac`

If a<0, the vertex is a maximum point, that is, the maximum value happens at
`x_(v)`, and it's value is
`y_(v)`.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

`ax^(2) + bx + c, a\\eq0`.

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

`x_(1) = (-b + √(\Delta))/(2*a)`

`x_(2) = (-b - √(\Delta))/(2*a)`

`\Delta= b^(2) - 4ac`

Quadratic equation:

`ax^2 + bx + c = 0`

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

This means that:

`-(b)/(2a) = 1.5`

`y_(v) = -(b^2-4ac)/(4a) = 676`

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

This means that:

`64a + 8b + c = 0`

Relation of b and a:

`-(b)/(2a) = 1.5`

This means that:

`b = -3a`

Relationship of c and a:

`64a + 8b + c = 0`

`64a - 24a + c = 0`

`c = -40a`

Finding a:

`y_(v) = -((-3a)^2-4a(-40a))/(4a) = 676`

`9a^2+160a = -2704a`

`9a^2 + 2864a = 0`

`a(9a + 2864) = 0`

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

`9a + 2864 = 0`

`9a = -2864`

`a = -(2864)/(9)`

`a = -318.22`

Finding b and c:

`b = -3a = -3(-318.22) = 954.67`

`c = -40a = -40(-318.22) = 12728.89`

The quadratic equation is given by:

`h(t) = -318.22t^2 + 954.67t + 12728.89`

The quadratic equation is
`h(t) = -318.22t^2 + 954.67t + 12728.89`