Question 195679
The initial velocity is 116 so {{{v = 116}}}



The initial height is 101 so {{{c = 101}}}



------------------------------------------------------------------------------------------------------------------------------


a)



{{{h=-16t^2+vt+c}}} Start with the given equation.



{{{h=-16t^2+116t+101}}} Plug in {{{v=116}}} and {{{c=101}}}



{{{0=-16t^2+116t+101}}} Plug in {{{h=0}}} (i.e., replace h with 0)


------------------------------------------------------------------------------------------------------------------------------

b)



Refer to part A above.



Notice we have a quadratic equation in the form of {{{0=at^2+bt+c}}} where {{{a=-16}}}, {{{b=116}}}, and {{{c=101}}}



Let's use the quadratic formula to solve for t 



{{{t = (-(116) +- sqrt( (116)^2-4(-16)(101) ))/(2(-16))}}} Plug in  {{{a=-16}}}, {{{b=116}}}, and {{{c=101}}}



{{{t = (-116 +- sqrt( 13456-4(-16)(101) ))/(2(-16))}}} Square {{{116}}} to get {{{13456}}}. 



{{{t = (-116 +- sqrt( 13456--6464 ))/(2(-16))}}} Multiply {{{4(-16)(101)}}} to get {{{-6464}}}



{{{t = (-116 +- sqrt( 13456+6464 ))/(2(-16))}}} Rewrite {{{sqrt(13456--6464)}}} as {{{sqrt(13456+6464)}}}



{{{t = (-116 +- sqrt( 19920 ))/(2(-16))}}} Add {{{13456}}} to {{{6464}}} to get {{{19920}}}



{{{t = (-116 +- sqrt( 19920 ))/(-32)}}} Multiply {{{2}}} and {{{-16}}} to get {{{-32}}}. 



{{{t = (-116 +- 4*sqrt(1245))/(-32)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{t = (-116+4*sqrt(1245))/(-32)}}} or {{{t = (-116-4*sqrt(1245))/(-32)}}} Break up the expression.  



{{{t = (29-sqrt(1245))/(8)}}} or {{{t = (29+sqrt(1245))/(8)}}} Reduce



So the exact roots are {{{t = (29-sqrt(1245))/(8)}}} or {{{t = (29+sqrt(1245))/(8)}}}



which approximate to {{{t=-0.78556969109433}}} or {{{t=8.03556969109433}}}



Since a negative time value doesn't make sense, the only practical root is approximately {{{t=8.03556969109433}}}



This rounds to one decimal place to get {{{t = "8.0"}}}



So it takes about 8.0 seconds for the rocket to hit the ground.