Question 1080606
Plot profit, {{{p}}} on the vertical axis and
selling price of a dress, {{{ d }}} on the horizontal axis
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If the relation is quadratic, then the equation
must look like:
{{{ p = a*d^2 + b*d + c }}}
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You are given 2 points:
( 110, 2300 ) ( the maximum {{{p}}} )
( 90, 1600 )
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For the maximum,
{{{ d[max] = -b/(2a) }}}
{{{ 110 = -b/(2a) }}}
{{{ 220a  = -b }}}
(1) {{{ b = -220a }}}
and
{{{ 2300 = a*110^2 + b*110 + c }}}
(2) {{{ 12100a + 110b + c = 2300 }}}
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( 90, 1600 )
{{{ 1600 = a*90^2 + b*90 + c }}}
(3) {{{ 8100a + 90b + c = 1600 }}}
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Subtract (3) from (2)
(2) {{{ 12100a + 110b + c = 2300 }}}
(3) {{{ -8100a - 90b - c = -1600 }}}
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{{{ 4000a + 20b = 700 }}}
{{{ 200a + b = 35 }}}
Plug (1) into this result
{{{ 200a - 220a = 35 }}}
{{{ 20a = -35 }}}
{{{ a = -7/4 }}}
and
(1) {{{ b = -220a }}}
(1) {{{ b = -220*( -7/4 ) }}}
(1) {{{ b = 385 }}}
and
(2) {{{ 12100a + 110b + c = 2300 }}}
(2) {{{ 12100*(-7/4) + 110*385 + c = 2300 }}}
(2) {{{ -21175 + 42350 + c = 2300 }}}
(2) {{{ 21175+ c = 2300 }}}
(2) {{{ c = -18875 }}}
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The equation is:
{{{ p = (-7/4)*d^2 + 385d - 18875 }}}
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check:
{{{ d[max] = -b/(2a) }}}
{{{ d[max] = -385/(2*(-7/4)) }}}
{{{ d[max] = -385/(-7/2) }}}
{{{ d[max] = 770/7 }}}
{{{ d[max] = 110 }}} ( as it should be )
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Plug this result into equation
{{{ p[max] = (-7/4)*110^2 + 385*110 - 18875 }}}
{{{ p[max] = (-7/4)*12100 + 42350 - 18875 }}}
{{{ p[max] = -21175 + 42350 - 18875 }}}
{{{ p[max] = 21175 - 18875 }}}
{{{ p[max] = 2300 }}} ( as it should be )
You can check the other point, ( 90, 1600 )