Question 856622
{{{ C(x) = .1x^2 - 60x + 22744 }}}
This is a parabola which has a minimum
Here's the plot:
{{{ C(x) }}} is plotted on the y-axis and
is rescaled to thousands of dollars
{{{ graph( 500, 500, -100, 700, -5, 30, .0001x^2 - .06x + 22.744 ) }}}
An initial guess would be {{{ C[min] = 14000 }}}
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When the form is:
{{{ a*x^2 + b*x + c }}}, the min ( or max ) is at
{{{ x[min] = -b/(2a) }}}
{{{ a = .1 }}}
{{{ b = -60 }}}
{{{ -b/(2a) = -(-60) / (2*.1) }}}
{{{ -b/(2a) = 60/.2 }}}
{{{ x[min] = 300 }}}
Now plug this value of {{{x}}} back into equation:
{{{ C(x) = .1x^2 - 60x + 22744 }}}
{{{ C(300) = .1*300^2 - 60*300 + 22744 }}}
{{{ C(300) = 9000 - 18000 + 22744 }}}
{{{ C(300) = 13744 }}}
The minimum cost is $13,744