Question 1042613
When the form of the equation is:
{{{ f(x) = a*x^2 + b*x + c }}}, then the
formula for the x-value of the vertex
( a peak in this case ) is:
{{{ x[max] = -b/(2a) }}}
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{{{ p(v) = -.004v^2 + 1.32v + 200 }}}
{{{ a = -.004 }}}
{{{ b = 1.32 }}}
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{{{ v[max] = -1.32 / ( 2*(-.004 )) }}}
{{{ v[max] = 1.32/.008 }}}
{{{ v[max] = 165 }}}
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She needs to sell 165 hot dogs/day to
maximize profit
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check:
Plug this value back into the equation to
get the maximum profit, {{{ p(v)[max] }}}
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{{{ p(v)[max] = -.004*165^2 + 1.32*165 + 200 }}}
{{{ p(v)[max] = -.004*27225 + 1.32*165 + 200 }}}
{{{ p(v)[max] = -108.9 + 217.8 + 200 }}}
{{{ p(v)[max] = 108.9 + 200 }}}
{{{ p(v)[max] = 308.9 }}}
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Here's the plot:
{{{ graph( 600, 400, -60, 500, -30, 350, -.004x^2 + 1.32x + 200 ) }}}