Question 1153135
Let {{{ n }}} = the number of $1 increases in ticket price
Let {{{ R }}} = total revenue from the tickets
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{{{ R = ( 300 - 5n )*( 4 + 1*n ) }}}
{{{ R = 1200 - 20n + 300n - 5n^2 }}}
{{{ R = -5n^2 + 280n + 1200 }}}
{{{ R =- n^2 + 56n + 240 }}}
The formula for {{{ n[max] }}} is
{{{ n[max] = -b/(2a) }}}
{{{ n[max] = -56/( 2*(-1) ) }}}
{{{ n[max] = 28 }}}
Plug this result back into equation to get {{{ R[max] }}}
{{{ R[max] = ( 300 - 5*28 )*( 4 + 28 ) }}}
{{{ R[max] = 160*32 }}}
{{{ R[max] = 5120 }}}
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The ticket price that gives this maximum revenue is:
{{{ 4 + 1*28 = 32 }}}
$32 per ticket
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check:
Here's the plot:
{{{ graph( 400, 400, -10, 65, -500, 5500, -5x^2 + 280x + 1200 ) }}}