Question 1040512
{{{ P(x) = -.002x^2 + 4.1x - 1600 }}}
This is a parabola with a maximum.
(a)
{{{ x = 700 }}}
{{{ P(700) = -.002*700^2 + 4.1*700 - 1600 }}}
{{{ P(700) = -.002*490000 + 2870 - 1600 }}}
{{{ P(700) = -980 + 2870 - 1600 }}}
{{{ P(700) = 290 }}}
$290 is the profit when 700 donuts are made
-----------------
(b)
The formula for the x-value of the maximum is:
{{{ x[max] = -b/(2a) }}}
{{{ x[max] = -4.1/(2*(-.002)) }}}
{{{ x[max] = 4.1/.004 }}}
{{{ x[max] = 1025 }}}
To find {{{ P[max] }}}, plug {{{ x[max] }}} back
into the equation
{{{ P(1025) = -.002*1025^2 + 4.1*1025 - 1600 }}}
{{{ P(1025) = -.002*1050625 + 4202.25 - 1600 }}}
{{{ P(1025) = -2101.25 + 4202.25 - 1600 }}}
{{{ P(1025) = 4202.5 - 3701.25 }}}
{{{ P(1025) = 501.25 }}}
The maximum profit, making 1025 donuts is $501.25
------------------
Here's the plot:
{{{ graph( 800, 400, -400, 1800, -50, 560, -.002x^2 + 4.1x - 1600 ) }}}