Question 765848


{{{P = -x^2 + 56x + 77}}}........this is a parabola; since the coefficient of leading term is {{{-1}}}, parabola turns downward and the vertex will be maximum

so, we need to find the coordinates ({{{h}}},{{{k}}}) of a vertex

use the formula {{{h = -b/2a}}} to find the {{{x}}} coordinate {{{h}}} of the vertex of a parabola and {{{k = P(h)}}} to find the {{{y}}} coordinate {{{k}}} of the vertex


{{{h = -56/2(-1)}}}

{{{h = -56/-2}}}

{{{h = 28}}}...........since {{{h = x}}}, this is the number of hot dogs

{{{P(28) = -28^2 + 56*28 + 77}}}

{{{P(28) =-784 + 1568 + 77}}}

{{{P(28) =861}}}..........since {{{k = y}}}, this is maximum profit

so, the vertex is at ({{{h}}},{{{k}}})=({{{28}}},{{{861}}})