Question 1031235
{{{ C(x) = x^2 - 140x + 5700 }}}
(a)
When the equation has the form:
{{{ y = a*x^2 + b*x + c }}} , then the
x-value of the vertex ( max or min ) is:
{{{ x[min] = -b/(2a) }}}
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In this case:
{{{ a = 1 }}}
{{{ b = -140 }}}
{{{ x[min] = -(-140) / (2*1) }}}
{{{ x[min] = 70 }}}
70 items should be produced to make
cost a minimum
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(b)
Plug {{{ x[min] }}} into equation to get {{{ C[min] }}}
{{{ C[min] = 70^2 - 140*70 + 5700 }}}
{{{ C[min] = 4900 - 9800 + 5700 }}}
{{{ C[min] = 800 }}}
The minimum cost is $800
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When does {{{ C(x) = 0 }}}?
{{{ x^2  - 140x + 5700 = 0 }}}
{{{ x^2 - 140x = -5700 }}}
complete the square
{{{ x^2 - 140x + ( -140/2 )^2 = -5700 + ( -140/2 )^2 }}}
{{{ x^2 - 140x + 4900 = -5700 + 4900 }}}
{{{ ( x - 70 )^2 = -800 }}}
I can't take the square root of
both sides and get a real answer,
so the cost can never be zero
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Here's the plot of {{{ C(x) }}}:
{{{ graph( 400, 400, -20, 140, -240, 2400, x^2 - 140x + 5700 ) }}}