Given:
Consider the height of the rocket, in feet after x seconds of launch is
[tex]y=-16x^2+152x+74[/tex]
To find:
The time at which the rocket will reach its max, to the nearest 100th of a second.
Solution:
We have,
[tex]y=-16x^2+152x+74[/tex]
It is a quadratic polynomial with negative leading coefficient. So, it is a downward parabola.
Vertex of a downward parabola is the point of maxima.
To find the time at which the rocket will reach its max, we need to find the x-coordinate of the vertex.
If a quadratic function is [tex]f(x)=ax^2+bx+c[/tex], then the vertex is
[tex]Vertex=\left(-\dfrac{b}{2a},f\left(-\dfrac{b}{2a}\right)\right)[/tex]
Here, [tex]a=-16,b=152,c=74[/tex].
So,
[tex]-\dfrac{b}{2a}=-\dfrac{152}{2(-16)}[/tex]
[tex]-\dfrac{b}{2a}=-\dfrac{152}{-32}[/tex]
[tex]-\dfrac{b}{2a}=4.75[/tex]
So, x-coordinate of the vertex is 4.75.
Therefore, the rocket will reach its max at 4.75 second.
