function word problem quadratics

Question

 

You will build a rectangular sheep enclosure next to a river. There is no need to build a fence along the river, so you only need to build on three sides. You have a total of feet of fence to use. Find the dimensions of the pen such that you can enclose the maximum possible area. One approach is to let  represent the length of fencing that runs perpendicular to the river, and write a formula for a function of  that outputs the area of the enclosure. Then find its vertex and interpret it.

Answer 1

let 470= 2 x+ y and A= x y then A= x(470 -2 x ) = 470 x -2 x² A= 470 x - 2 x² dA/dx =470- 4 x and d²A/dx² = -4 let dA/dx =0 to maximize the equation so 0= 470 - 4 x and x =117.5 ft this is a maximum since the second derivative is negative and y=245 ft.