Question 196819
Is the given equation {{{P=4t+sqrt(t+110)}}} ???


{{{P=4t+sqrt(t+110)}}} Start with the given equation.



{{{500=4t+sqrt(t+110)}}} Plug in {{{P=500}}}



{{{500-4t=sqrt(t+110)}}} Subtract 4t from both sides.



{{{(500-4t)^2=(sqrt(t+110))^2}}} Square both sides



{{{(500-4t)^2=t+110}}} Square the square root to eliminate it.



{{{250000-4000t+16t^2=t+110}}} FOIL



{{{250000-4000t+16t^2-t-110=0}}} Get all terms to the left side.



{{{16t^2-4001t+249890=0}}} Combine like terms.



Notice we have a quadratic in the form of {{{At^2+Bt+C}}} where {{{A=16}}}, {{{B=-4001}}}, and {{{C=249890}}}



Let's use the quadratic formula to solve for t



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



{{{t = (-(-4001) +- sqrt( (-4001)^2-4(16)(249890) ))/(2(16))}}} Plug in  {{{A=16}}}, {{{B=-4001}}}, and {{{C=249890}}}



{{{t = (4001 +- sqrt( (-4001)^2-4(16)(249890) ))/(2(16))}}} Negate {{{-4001}}} to get {{{4001}}}. 



{{{t = (4001 +- sqrt( 16008001-4(16)(249890) ))/(2(16))}}} Square {{{-4001}}} to get {{{16008001}}}. 



{{{t = (4001 +- sqrt( 16008001-15992960 ))/(2(16))}}} Multiply {{{4(16)(249890)}}} to get {{{15992960}}}



{{{t = (4001 +- sqrt( 15041 ))/(2(16))}}} Subtract {{{15992960}}} from {{{16008001}}} to get {{{15041}}}



{{{t = (4001 +- sqrt( 15041 ))/(32)}}} Multiply {{{2}}} and {{{16}}} to get {{{32}}}. 



{{{t = (4001 +- 13*sqrt(89))/(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 = (4001+13*sqrt(89))/(32)}}} or {{{t = (4001-13*sqrt(89))/(32)}}} Break up the expression.  



{{{t=128.864}}} or {{{t=121.199}}} Use a calculator to approximate



Now because plugging in {{{t=128.864}}} back in the second equation results in a contradiction, this means that {{{t=128.864}}} is NOT a solution.



So the only solution is approximately {{{t=121.199}}}



This means that it will take about 122 days (rounding up to the nearest whole number) for the fish population to reach 500.