Question 119148
Do you want to solve for t?




{{{77=-16t^2+80t+5}}} Start with the given equation



{{{0=-16t^2+80t+5-77}}}  Subtract 77 from both sides. 



{{{0=-16t^2+80t-72}}} Combine like terms



Let's use the quadratic formula to solve for t:



Starting with the general quadratic


{{{at^2+bt+c=0}}}


the general solution using the quadratic equation is:


{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{-16*t^2+80*t-72=0}}} ( notice {{{a=-16}}}, {{{b=80}}}, and {{{c=-72}}})





{{{t = (-80 +- sqrt( (80)^2-4*-16*-72 ))/(2*-16)}}} Plug in a=-16, b=80, and c=-72




{{{t = (-80 +- sqrt( 6400-4*-16*-72 ))/(2*-16)}}} Square 80 to get 6400  




{{{t = (-80 +- sqrt( 6400+-4608 ))/(2*-16)}}} Multiply {{{-4*-72*-16}}} to get {{{-4608}}}




{{{t = (-80 +- sqrt( 1792 ))/(2*-16)}}} Combine like terms in the radicand (everything under the square root)




{{{t = (-80 +- 16*sqrt(7))/(2*-16)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{t = (-80 +- 16*sqrt(7))/-32}}} Multiply 2 and -16 to get -32


So now the expression breaks down into two parts


{{{t = (-80 + 16*sqrt(7))/-32}}} or {{{t = (-80 - 16*sqrt(7))/-32}}}



Now break up the fraction



{{{t=-80/-32+16*sqrt(7)/-32}}} or {{{t=-80/-32-16*sqrt(7)/-32}}}



Simplify



{{{t=5 / 2-sqrt(7)/2}}} or {{{t=5 / 2+sqrt(7)/2}}}



So these expressions approximate to


{{{t=1.1771243444677}}} or {{{t=3.8228756555323}}}



So our solutions are:

{{{t=1.1771243444677}}} or {{{t=3.8228756555323}}}





So when the time is {{{t=1.1771243444677}}} or {{{t=3.8228756555323}}} the ball is  at a height of 77 ft.