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Something's Fishy
The Survivors are quickly running out of food and decide to build some fish traps using netting. The locals on the island usually put 5 traps into the water, and catch 46 sardines (really, really small fish) in each trap. They have found that for each additional trap they put in the water, they catch 2 less sardines per trap.
The above situation can be represented by the equation $N = - 2 x^{2} + 56 x$ , where $N$ represent the number of sardines caught, and $x$ , the number of traps. Rewrite the equation in factored form and vertex form:
State the optimal mumber of traps to use, and the total mumber of sardines they will catch

Answer

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Answer

$Optimal number of traps: 14, Total number of sardines: 392$

Steps

Step 1 ：Rewrite the equation in factored form: $N = - 2 x (x - 28)$

Step 2 ：Find the vertex of the parabola: $x = \frac{- b}{2 a} = \frac{- 56}{2 (- 2)} = 14$

Step 3 ：Calculate the number of sardines at the vertex: $N = - 2 (14) (14 - 28) = 392$

Step 4 ：Rewrite the equation in vertex form: $N = 392 - 392 (0.0714 x - 1)^{2}$

Step 5 ： $Optimal number of traps: 14, Total number of sardines: 392$