Question 1060581
Let {{{ n }}} = the number of $1 decreases in price
Let {{{ I }}} = the total income from sales of water pails
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[ income ] = [ price per pail ] x [ number of pails sold ]
{{{ I = ( 20 - 1*n )*( 80 + 15n ) }}}
{{{ I = 1600 - 80n + 300n - 15n^2 }}}
{{{ I = -15n^2 + 220n + 1600 }}}
The n-value of the vertex ( max in this case ) is:
{{{ n[v] = -b/(2a) }}}
{{{ a = -15 }}}
{{{ b = 220 }}}
{{{ n[v] = -220/( 2*(-15) ) }}}
{{{ n[v] = 7.333 }}}
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Going back to original equation:
[ price/pail ] = {{{ 20 - n }}}
The price that maximizes income is:
{{{ 20 - 7.333 = 12.333 }}}
$12.33
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The max income at this price is:
{{{ I = -15n^2 + 220n + 1600 }}}
{{{ I[max] = -15*7.333^2 + 220*7.333 + 1600 }}}
{{{ I[max] = -806.593 + 1613.26 + 1600 }}}
{{{ I[max] = 2406.67 }}}
$2,406.67 
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here's the plot:
{{{ graph( 600, 400, -2, 25, -300, 2600, -15x^2 + 220x + 1600 ) }}}