Question 1094144
Let {{{ P }}} = profit per week
Let {{{ n }}} = the number of $.50 decreases in price
[ profit ] = [ income ] - [ cost ]
{{{ P = ( 40 - .5n )*( 400 + 20n ) - ( 400 + 20n )*5 }}}
{{{ P = ( 40 - 5 - .5n )*( 400 + 20n ) }}}
{{{ P = ( 35 - .5n )*( 400 + 20n ) }}}
{{{ P = 14000 - 200n + 700n - 10n^2 }}}
{{{ P = -10n^2 + 500n + 14000 }}}
{{{ P = -n^2 + 50n + 1400 }}}
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The n-value of the maximum is at:
{{{ n[max] = -b/(2a) }}}
{{{ a = -1 }}}
{{{ b = 50 }}}
{{{ n[max] = -50/(2*(-1)) }}}
{{{ n[max] = 25 }}}
and
{{{ P[max] = -10*25^2 + 500*25 + 14000 }}}
{{{ P[max] = -6250 + 12500 + 14000 }}}
{{{ P[max] = 20250 }}}
The maximum weekly profit is $20,250
The price that will maximize the profit is:
{{{ 40 - .5n }}}
{{{ 40 - .5*25 }}}
{{{ 40 - 12.5 = 27.5 }}}
$27.50 per doll
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check:
Here's the plot:
{{{ graph( 400, 400, -10, 60, -2000, 25000, -10x^2 + 500x + 14000 ) }}}