The Jacksons wish to build a rectangular enclosure for their dog, Fido. They purchased 100 feet of fencing for the enclosure. The area of the enclosure, A, is related to the length of one of the sides, x (measured in feet), by the formula f (x) = - x^2 + 50 x. Identify the vertex of this situation and explain its meaning in the context of the problem?

The vertex can be found by completing the square:

f (x) = - (x^2 - 50x)

= - (x^2 + 625 - 625 - 50x) (take half the coeff of x, square

= - (x^2 + 625 - 50x) + 625 it then add and subtract it)

= - (x^2 - 25x - 25x + 625) + 625

= - (x (x - 25) - 25 (x + 25)) + 625

= - ((x - 25) (x + 25)) + 625

= - ((x + 25) ^2 + 625

so the vertex is (25, 625)

in this context this is the 'optimized' area or the largest possible area with the given perimeter (so it is basically a square, 25 x 4 = 100)