Question 732968
The curve that the rocket follows is a parabola. The maximum
height is also called the vertex of the parabola. The formula for
the time component of the vertex is {{{ t[v] = -b/(2a) }}}, when
the equation looks like {{{ f(t) = at^2 + b*t + c }}}
In this problem,
{{{ a = -16 }}}
{{{ b = 72 }}}
{{{ t[v] = -72 / (2*(-16)) }}}
{{{ t[v] = 72/32 }}}
{{{ t[v] = 2.25 }}} sec
Now plug this value back into the equation to find {{{ f(2.25) }}}
{{{ f(2.25) = -16*(2.25)^2 + 72*2.25 + 220 }}}
{{{ f(2.25) = -16*5.0625 + 162 + 220 }}}
{{{ f(2.25) = -81 + 162 + 220 }}}
{{{ f(2.25) = 301 }}}
It reaches a maximum height of 301 ft in 2.25 sec
Here's a plot:
{{{ graph( 400, 400, -1, 8, -20, 330, -16x^2 + 72x + 220 ) }}}