Question 452984
Profit is (revenue - cost)
{{{R = 100x - .5x^2
{{{C = 60x + 300 }}}
{{{ R - C = 100x - .5x^2 - 60x - 300 }}}
{{{ P = - .5x^2 + 40x - 300 }}}
{{{ 300 = - .5x^2 + 40x - 300 }}}
{{{ - .5x^2 + 40x - 600 = 0 }}}
{{{ - 5x^2 + 400x - 6000 = 0 }}}
{{{ - x^2 + 80x - 1200 = 0 }}}
{{{ x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ a = -1 }}}
{{{ b = 80 }}}
{{{ c = -1200 }}}
{{{x = (-80 +- sqrt( 80^2 - 4*(-1)*(-1200) ))/(2*(-1)) }}}
{{{x = (-80 +- sqrt( 6400 - 4800 ))/((-2)) }}}
{{{x = (-80 +- sqrt( 1600 ))/((-2)) }}}
{{{x = (-80 + 40)/((-2)) }}}
{{{ x = 20 }}}
and, using the negative root
{{{x = (-80 - 40)/((-2)) }}}
{{{ x = 60 }}}
Selling either 20 or 60 items per week will
give a profit of $300
Here's a graph of solutions and the profit curve:
{{{ graph( 400, 400, -50, 70, -50, 500, -x^2 + 80x - 1200) }}}