Question 1170724
a. How long will the flare be {{{15m}}} high or higher?

{{{h(t)>=15}}}

{{{ -5t^2+30t >=15}}}.....simplify
{{{ -t^2+6t >=3}}}
{{{ -t^2+6t-3 >=0}}}
{{{ t>=(-b+-sqrt(b^2-4ac))/2a}}}
{{{ t>=(-6+-sqrt(6^2-4(-1)(-3)))/-2}}}
{{{ t>=(-6+-sqrt(24))/-2}}}
{{{ t>=(-6+-4.9)/-2}}}

need only positive solution
{{{ t>=(-6-4.9)/-2}}}
{{{ t>=5.5}}}seconds



b.When will the flare hit the water?

{{{h(t)=0}}}

{{{ -5t^2+30t=0}}}
{{{ -t^2+6t=0}}}
{{{ -t(t-6)=0}}}
need only positive solution
{{{t=6}}} seconds


c. How high will the flare be after 2 seconds?
{{{t=2}}}

{{{h(2)=-5*2^2+30*2}}}
{{{ h(2)=-20+60}}}
{{{ h(2)=40}}}



d.How high will the flare be after {{{5 }}}seconds?

{{{t=5}}}

{{{ h(5)=-5*5^2+30*5}}}
{{{ h(5)=-125+150}}}
{{{ h(5)=25}}}

e.What is the maximum height the flare will reach? 

the maximum height will be at vertex, write equation in vertex form

{{{h(t)=-5t^2+30t}}}......complete square
{{{h(t)=-5(t^2-6t)}}}
{{{h(t)=-5(t^2-6t+b^2)-(-5b^2)}}}
{{{h(t)=-5(t^2-6t+3^2)+5*3^2}}}
{{{h(t)=-5(t-3)^2+45}}}

=> vertex is at ({{{3}}},{{{45}}})

the maximum height will be {{{45}}}meters


f. How long will it take for that max to occur?

the maximum height the flare will reach in {{{3}}} seconds