Question 1031134
With zero increase in ticket price, The 
number of tickets sold is:
{{{ n = 500 }}} and income is:
{{{ 15*500 = 7500 }}}
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When there are {{{ 1*n }}} dollar increases
in the ticket price, the income is:
{{{ I = ( 15 + n )*( 500 - 20n ) }}}
{{{ I = 7500 + 500n - 300n - 20n^2 }}}
{{{ I = -20n^2 + 200n + 7500 }}}
This function is a maximum when:
{{{ n[max] = -b/(2a) }}}, where
{{{ a = -20 }}}
{{{ b = 200 }}}
{{{ n[max] = -200 / ( 2*(-20) ) }}}
{{{ n[max] = 5 }}}
and
{{{ I = -20*5^2 + 200*5 + 7500 }}}
{{{ I = -500 + 1000 + 7500 }}}
{{{ I = 8000 }}}
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With {{{ n = 5 }}} $1 increases in ticket price,
the price that will maximize profit is:
{{{ 15 + n = 15 + 5 }}}
{{{ 15 + n = 20 }}}
$20
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Here's the plot of income:
{{{ graph( 400, 400, -10, 30, -1000, 9000, -20x^2 + 200x + 7500 ) }}}