Question 1078618
The income per night is:
{{{ x*( -.02x + 9 ) }}}
The cost to make pizza per night is:
{{{ 3x + 110 }}}
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Let {{{ P }}} = profit per night
[ profit ] = [ income ] - [ cost ]
{{{ P = x*( -.02x + 9 ) - 3x - 110 }}}
{{{ P = -.02x^2 + 9x - 3x - 110 }}}
{{{ P = -.02x^2 + 6x - 110 }}}
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The x-value of the maximum is:
{{{ x[max] = -b/(2a) }}}
{{{ a = -.02 }}}
{{{ b = 6 }}}
{{{ -b/(2a) = -6/(2*(-.02)) }}}
{{{ x[max] = -6/(-.04) }}}
{{{ x[max] = 150 }}}
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plug this back into {{{ P }}} to get {{{ P[max] }}}
{{{ P[max] = -.02*150^2 + 6*150 - 110 }}}
{{{ P[max] = -450 + 900 - 110 }}}
{{{ P[max] = 340 }}}
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The maximum profit of $340/night can be
earned by making 150 pizzas
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Here's a plot of the profit function:
{{{ graph( 400, 400, -40, 350, -40, 400, -.02x^2 + 6x - 110 ) }}}