Question 196647
Since the "total time is 2.3 s", this means that {{{t[1]+t[2]=2.3}}}



{{{t[1]+t[2]=sqrt(d)/4+d/1090}}} Start with the given equation.



{{{2.3=sqrt(d)/4+d/1090}}} Plug in {{{t[1]+t[2]=2.3}}}



{{{2.3(2180)=(sqrt(d)/cross(4))cross(2180)^(545)+(d/cross(1090))cross(2180)^(2)}}} Multiply EVERY term by the LCD {{{2180}}} to clear out the fractions.



{{{5014=545sqrt(d)+2d}}} Multiply and simplify



{{{5014-2d=545sqrt(d)}}} Subtract 2d from both sides.



{{{(5014-2d)^2=(545sqrt(d))^2}}} Square both sides



{{{(5014-2d)^2=297025d}}} Square {{{545sqrt(d)}}} to get {{{297025d}}}



{{{25140196-20056d+4d^2=297025d}}} FOIL the left side



{{{25140196-20056d+4d^2-297025d=0}}} Get all terms to the left side.



{{{4d^2-317081d+25140196=0}}} Combine like terms.



Notice we have a quadratic in the form of {{{Ad^2+Bd+C}}} where {{{A=4}}}, {{{B=-317081}}}, and {{{C=25140196}}}



Let's use the quadratic formula to solve for d



{{{d = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{d = (-(-317081) +- sqrt( (-317081)^2-4(4)(25140196) ))/(2(4))}}} Plug in  {{{A=4}}}, {{{B=-317081}}}, and {{{C=25140196}}}



{{{d = (317081 +- sqrt( (-317081)^2-4(4)(25140196) ))/(2(4))}}} Negate {{{-317081}}} to get {{{317081}}}. 



{{{d = (317081 +- sqrt( 100540360561-4(4)(25140196) ))/(2(4))}}} Square {{{-317081}}} to get {{{100540360561}}}. 



{{{d = (317081 +- sqrt( 100540360561-402243136 ))/(2(4))}}} Multiply {{{4(4)(25140196)}}} to get {{{402243136}}}



{{{d = (317081 +- sqrt( 100138117425 ))/(2(4))}}} Subtract {{{402243136}}} from {{{100540360561}}} to get {{{100138117425}}}



{{{d = (317081 +- sqrt( 100138117425 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{d = (317081 +- 545*sqrt(337137))/(8)}}} 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>)  



{{{d = (317081+545*sqrt(337137))/(8)}}} or {{{d = (317081-545*sqrt(337137))/(8)}}} Break up the expression.  



So the answers are {{{d = (317081+545*sqrt(337137))/(8)}}} or {{{d = (317081-545*sqrt(337137))/(8)}}} 



which approximate to {{{d=79190.884}}} or {{{d=79.366}}} 



Notice that if the time for the stone to fall was 2.3 seconds, then the depth would be AT MOST {{{d=16(2.3)^2=84.64}}}. In other words, the MAX depth that is possible in this problem is about 85 feet. 



Since the value {{{d=79190.884}}} is too large, this means that {{{d=79.366}}} is the only answer.



So to the nearest whole number, the depth of the well is about 79 feet.