Question 190294
Well, first off, the equation should be {{{-4.9t^2+245t=1960}}} (let me know if you need help figuring out the equation)



{{{-4.9t^2+245t=1960}}} Start with the given equation.



{{{-4.9t^2+245t-1960=0}}} Subtract 1960 from both sides.



{{{-49t^2+2450t-19600=0}}} Multiply EVERY term by 10 to make every number a whole number.



Notice we have a quadratic equation in the form of {{{at^2+bt+c}}} where {{{a=-49}}}, {{{b=2450}}}, and {{{c=-19600}}}



Let's use the quadratic formula to solve for t



{{{t = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{t = (-(2450) +- sqrt( (2450)^2-4(-49)(-19600) ))/(2(-49))}}} Plug in  {{{a=-49}}}, {{{b=2450}}}, and {{{c=-19600}}}



{{{t = (-2450 +- sqrt( 6002500-4(-49)(-19600) ))/(2(-49))}}} Square {{{2450}}} to get {{{6002500}}}. 



{{{t = (-2450 +- sqrt( 6002500-3841600 ))/(2(-49))}}} Multiply {{{4(-49)(-19600)}}} to get {{{3841600}}}



{{{t = (-2450 +- sqrt( 2160900 ))/(2(-49))}}} Subtract {{{3841600}}} from {{{6002500}}} to get {{{2160900}}}



{{{t = (-2450 +- sqrt( 2160900 ))/(-98)}}} Multiply {{{2}}} and {{{-49}}} to get {{{-98}}}. 



{{{t = (-2450 +- 1470)/(-98)}}} Take the square root of {{{2160900}}} to get {{{1470}}}. 



{{{t = (-2450 + 1470)/(-98)}}} or {{{t = (-2450 - 1470)/(-98)}}} Break up the expression. 



{{{t = (-980)/(-98)}}} or {{{t =  (-3920)/(-98)}}} Combine like terms. 



{{{t = 10}}} or {{{t = 40}}} Simplify. 



So the answers are {{{t = 10}}} or {{{t = 40}}} 



This means that at 10 seconds the flare is at the height of 1960 m. Also, the flare is 1960 m high when the time is 40 seconds. 



So the difference between these two times is 40-10=30 seconds (think of t=10 as time zero for the balloonist)



So it takes 30 seconds for the flare to come back down to the height of 1960 m.