Question 1141780
Let {{{ n }}} = the number of $1 increases in the ticket price
Their prediction is that {{{ 600 - 25n }}} will be the number
of tickets they will sell at a price of {{{ 10 + n }}} dollars each
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Let {{{ R }}} = revenue from sales of tickets
{{{ R = ( 600 - 25n )*( 10 + n ) }}}
What value of {{{ n }}} gives them the maximum revenue?
{{{ R = 6000 - 250n + 600n - 25n^2 }}}
{{{ R = -25n^2 + 350n + 6000 }}}
The plot of {{{ R(n) }}} is a parabola with a maximum n-value at
{{{ -b/(2a) = -350/(2*(-25)) }}}
{{{ -b/(2a) = 7 }}}
When {{{ n = 7 }}}, revenue is a maximum
{{{ R{7} = ( 600 - 25*7 )*( 10 + 7 ) }}}
{{{ R(7) = ( 600 - 175 )*17 }}}
{{{ R[7] = 425*17 }}}
{{{ R(7) = 7225 }}}
$7,225 is the maximum revenue they can make.
Revenue will increase for each $1 increase in ticket price
until the tickets cost {{{ 10 + 7 = 17 }}} $17
After that, Revenue will decrease.
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Here is the plot of ( n, R )
{{{ graph( 600, 400, -2, 15, -1000, 10000, -25x^2 + 350x + 6000 ) }}}