Question 1015369
You are missing a minus sign in front of the {{{16t^2}}} .
{{{system(f(t)=-16t^2+V[0]t+H[0],V[0]=80,H[0]=5)}}}--->{{{f(t)=-16t^2+80t+5}}}
 
JUST USING ALGEBRA:
{{{f(t)=-(4t)^2+2*(4t)*10-10^2+5+10^2=-((4t)^2-2*(4t)*10+10^2)+5+100=-(4t-10)^2+105<=105}}} .
 
APPLYING A FORMULA:
{{{f(t)=-16t^2+80t+5}}} is a quadratic function.
A quadratic function {{{f(x)=ax^2+bx+c}}} with {{{a<0}}} has a maximum at {{{x=(-b)/2a}}} ,
so, in this case, with {{{system(x=t,a=-16,b=80,c=5)}}} ,
the maximum happens at {{{t=(-80)/(2(-16))=(-80)/(-32)=2.5}}} .
The maximum is {{{f(2.5)=-16*2.5^2+80*2.5+5=-16*6.25+200+5-100+200+5=105}}}
 
So, the maximum height of the water balloon is {{{highlight(105feet)}}} .