Answer:
Boulder's maximum height is 226 ft.
It take 3.5 seconds the Boulder to reach its maximum height.
Step-by-step explanation:
Given :
A catapult launches a boulder with an upward velocity of 112 ft/s.
The height of the Boulder, h, in feet
After t seconds is given by the function :
\(h = -16t^{2} +112t+30\) --(a)
A quadratic function can be graphed using a table of values. The graph creates a parabola.
If the coefficient of the squared term is positive then the parabola opens up and The vertex of this parabola is known as the minimum point.
If the coefficient of the squared term is negative then the parabola opens down and The vertex of this parabola is known as the maximum point.
Now we will use vertex formula to calculate t
When \(f(x)= ax^{2} +bx+c\)
then vertex will be
\(x=\frac{-b}{2a}\)
Now consider the given function:
\(h = -16t^{2} +112t+30\)
where a = -16
b=112
so using vertex formula :
\(t=\frac{-112}{2*(-16)}\)
\(t=3.5\)
Thus, it take 3.5 seconds the Boulder to reach its maximum height.
Now to calculate the Boulder's maximum height . Put value of t = 3.5 in given function (a)
\(h = -16(3.5)^{2} +112(3.5)+30\)
\(h = -196 +392+30\)
\(h = 226\)
Thus, Boulder's maximum height is 226 ft.
