Question 930432
a) The maximum for a quadratic function {{{f(x)=ax^2+bx+c}}}
happens at {{{x=-b/2a}}} .
In this case, {{{x=-24/(2*(-16))=24/32=3/4=0.75}}} ,
so it reaches maximum height at {{{highlight(0.75)}}} seconds.


b) At that point, the height in feet is
{{{-16*0.75^2+24*0.75+75=highlight(84)}}} feet.


c) {{{0=-16x^2+24x+75}}} is an equation I would solve by "completing the square".
{{{16x^2-24x=75}}}
{{{(4x)^2-2*3*(4x)=75}}}
{{{(4x)^2-2*3*(4x)+3^2=75+3^2}}}
{{{(4x-3)^2=75+9}}}
{{{(4x-3)^2=84}}}
Since the other solution is negative, the only solution that makes sense is:
{{{4x-3=sqrt(84)}}}
{{{4x-3=2sqrt(21)}}}
{{{4x=3+2sqrt(21)}}}
{{{x=(3+2sqrt(21))/4}}}
{{{highlight(x=3.04)}}} seconds (rounded).
 
USING THE QUADRATIC FORMULA, the calculation becomes a bit more painful:
The quadratic formula says that the solutions to {{{ax^2+bx+c=0}}} are given by
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In this case, with {{{a=-16}}} , {{{b=24}}} , and {{{c=75}}} ,
so {{{x = (-24 +- sqrt((-24)^2-4*(-16)*75))/(2*(-16)) }}}
{{{x = (-24 +- sqrt(576+4800))/(-32) }}}
{{{x = (-24 +- sqrt(5376))/(-32) }}}
Approximating, we get
{{{x = (-24 +- 73.3212)/(-32) }}}
The negative solution does not make sense.
The positive solution is
{{{x = (-24 - 73.3212)/(-32) }}}
{{{x = (-97.3212)/(-32) }}}
{{{x = 97.3212/32 }}}
{{{highlight(x=3.04)}}} seconds (rounded).