Question

A hiker stands on a small hill and tosses a stone. The stone then follows a path with height h above the ground at time t. The function

h(t) = - 16t2 + 16x + 32 models the stone's path. How long does it take the stone to reach its maximum height?

Answer 1

At time t=0.5, the stone reaches a maximum height of h= 35.

Explanation:

A quadratic function or function of the second degree is a polynomial function defined by:

f(x)=a*x² + b*x +c

The maximums and minimums of a function are the largest (maximum) or smallest (minimum) values ​​of a function.

The vertex of the function is a point that is part of the parabola or quadratic function that coincides with the maximum or minimum. When a> 0, the vertex of the parabola is at the bottom of it, being a minimum (that is, the parabola opens "upward"), and when a <0 the vertex is at the top , being a maximum (that is, the parable opens "downward").

h represents the value of its abscissa (value of x).

k represents the value of its ordinate (value of f (x)).

The value of h can be calculated with the formula
`h=-(b)/(2*a)`

The value of k must be obtained by substituting the value of h in the function f(x)=a*x² + b*x +c

In this case: a=-16, b=16 and c=32. As a <0 the vertex is at the top , being a maximum. So:

`h=-(16)/(2*(-16))=0.5`

Substituting the value of h in the function f(x)=-16*x² + 16*x +32 you get:

f(0.5)=-16*0.5² + 16*0.5 +32

f(0.5)= 35

At time t=0.5, the stone reaches a maximum height of h= 35.