SOLUTION: Hello! I've been having some trouble with the following word problem A large pond is stocked with fish. The fish population P is modeled by the formula P = 4t + 8√t + 1

Algebra ->  Equations -> SOLUTION: Hello! I've been having some trouble with the following word problem A large pond is stocked with fish. The fish population P is modeled by the formula P = 4t + 8√t + 1      Log On


   

Question 196819: Hello!
I've been having some trouble with the following word problem
A large pond is stocked with fish. The fish population P is modeled by the formula P = 4t + 8√t + 110, where t is the number of days since the fish were first introduced into the pond. How many days will it take for the fish population to reach 500?
thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Is the given equation P=4t%2Bsqrt%28t%2B110%29 ???

P=4t%2Bsqrt%28t%2B110%29 Start with the given equation.


500=4t%2Bsqrt%28t%2B110%29 Plug in P=500


500-4t=sqrt%28t%2B110%29 Subtract 4t from both sides.


%28500-4t%29%5E2=%28sqrt%28t%2B110%29%29%5E2 Square both sides


%28500-4t%29%5E2=t%2B110 Square the square root to eliminate it.


250000-4000t%2B16t%5E2=t%2B110 FOIL


250000-4000t%2B16t%5E2-t-110=0 Get all terms to the left side.


16t%5E2-4001t%2B249890=0 Combine like terms.


Notice we have a quadratic in the form of At%5E2%2BBt%2BC where A=16, B=-4001, and C=249890


Let's use the quadratic formula to solve for t


t+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


Plug in A=16, B=-4001, and C=249890


t+=+%284001+%2B-+sqrt%28+%28-4001%29%5E2-4%2816%29%28249890%29+%29%29%2F%282%2816%29%29 Negate -4001 to get 4001.


t+=+%284001+%2B-+sqrt%28+16008001-4%2816%29%28249890%29+%29%29%2F%282%2816%29%29 Square -4001 to get 16008001.


t+=+%284001+%2B-+sqrt%28+16008001-15992960+%29%29%2F%282%2816%29%29 Multiply 4%2816%29%28249890%29 to get 15992960


t+=+%284001+%2B-+sqrt%28+15041+%29%29%2F%282%2816%29%29 Subtract 15992960 from 16008001 to get 15041


t+=+%284001+%2B-+sqrt%28+15041+%29%29%2F%2832%29 Multiply 2 and 16 to get 32.


t+=+%284001+%2B-+13%2Asqrt%2889%29%29%2F%2832%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


t+=+%284001%2B13%2Asqrt%2889%29%29%2F%2832%29 or t+=+%284001-13%2Asqrt%2889%29%29%2F%2832%29 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.