Question 1028708
{{{ P(d) = -2d^2 + 13d - 6 }}}
The break even point is where {{{ P(d) = 0 }}},
or no profit
(a)
{{{ -2d^2 + 13d - 6 = 0 }}}
Use quadratic formula
{{{ d = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = -2 }}}
{{{ b = 13 }}}
{{{ c = -6 }}}
{{{ d = (-13 +- sqrt( 13^2 - 4*(-2)*(-6) )) / (2*(-2)) }}}
{{{ d = (-13 +- sqrt( 169 - 48 )) /(-4 )}}}
{{{ d = (-13 +- sqrt( 121 )) /(-4 )}}}
{{{ d = (-13 + 11) /(-4 )}}}
{{{ d = (-2) / (-4) }}}
{{{ d = .5 }}} hundreds
and
{{{ d = ( -13 - 11 ) / (-4) }}}
{{{ d = (-24) / (-4) }}}
{{{ d = 6 }}} hundreds
There are 2 break even points:
50 watches sold and
600 watches sold
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(b)
The maximum profit is where:
{{{ p = -b/(2a) }}}
{{{ p = -13/( 2*(-2)) }}}
{{{ p = (-13)/(-4) }}}
{{{ p = 13/4 }}}
{{{ (13/4)*100 = 325 }}}
When 325 watches are sold, the profit is maximum
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check:
here's the plot:
{{{ graph( 500, 500, -2,10, -10, 16, -2x^2 + 13x - 6 ) }}}