Problem
$-3$. 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
Solution
Step 1 ：Rewrite the equation in factored form: \(N = -2x(x - 28)\)
Step 2 ：Find the vertex of the parabola: \(x = \frac{-b}{2a} = \frac{-56}{2(-2)} = 14\)
Step 3 ：Calculate the number of sardines at the vertex: \(N = -2(14)(14 - 28) = 392\)
Step 4 ：Rewrite the equation in vertex form: \(N = 392 - 392(0.0714x - 1)^2\)
Step 5 ：\(\boxed{\text{Optimal number of traps: } 14, \text{Total number of sardines: } 392}\)
