Question 1062337
The solutions, or roots are {{{ x = 0 }}} and {{{ x = 50 }}}
and half-way between these points at {{{ x = 25 }}},
I have the point ( 25, 4.5 )
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(a)
The general equation would look like:
{{{ y = -a*x^2 + b*x }}}
( I need the squared term to be negative, since
that gives me a peak and not a dip )
The solutions are:
( 0,0 )
{{{ 0 = -a*0^2 + b*0 }}}
{{{ 0 = 0 }}}
and
( 50,0 )
{{{ 0 = -a*50^2 + b*50 }}}
{{{ 2500a = 50b }}}
(1) {{{ b = 50a }}}
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The peak, or vertex, is at ( 25, 4.5 ), so
{{{ 4.5 = -a*25^2 + b*25 }}}
(2) {{{ 625a = 25b - 4.5 }}}
Plug (1) into (2)
(2) {{{ 625a = 25*50a - 4.5 }}}
(2) {{{ 1250a - 625a = 4.5 }}}
(2) {{{ 625a = 4.5 }}}
(2) {{{ a = .0072 }}}
and
(1) {{{ b = 50a }}}
(1) {{{ b = 50*.0072 }}}
(1) {{{ b = .36 }}}
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So, the equation is:
{{{ y = -.0072x^2 + .36x }}}
( Note there is no constant term because the
parabola goes through ( 0,0 ) )
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(b)
{{{ 50 - 20 = 30 }}} m
There is {{{ 15 }}} m clearance on both sides
So, I have the points
( 15, y[1] ) and
( 35, y[2] ) as points on the bridge that are
directly above the ends of the floating platform
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( 15, y[1] )
{{{ y[1] = -.0072*15^2 + .36*15 }}}
{{{ y[1] = -.0072*225 + 5.4 }}}
{{{ y[1] = -1.62 + 5.4 }}}
{{{ y[1] = 3.78 }}} m
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( 35, y[2] )
{{{ y[2] = -.0072*35^2 + .36*35 }}}
{{{ y[2] = -.0072*1225 + 12.6 }}}
{{{ y[2] = -8.82 + 12.6 }}}
{{{ y[2] = 3.78 }}} m
( this makes sense because the platform is
exactly in the middle )
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If you want {{{ 20/100 = .2 }}} m on each side,
then what value of {{{ y }}} goes with {{{ x[1] = 15 - .2 }}}
or {{{ x[2] = 35 + .2 }}}
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{{{ y[3] = -.0072*14.8^2 + .36*14.8 }}}
{{{ y[3] = -.0072*219.04 + 5.328 }}}
{{{ y[3] = -1.577 + 5.328 }}}
{{{ y[3] = 3.751 }}} m
This is the height of the platform that gives me {{{ 20 }}} cm
clearance on the sides. 
The points on the platform are below the bridge, and they are at
( 15, 3.751 ) and
( 35, 3.751 )
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I would definitely get a 2nd opinion
I coiuld easily miscalculate
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Here's a plot of my equation:
{{{ graph( 400, 400, -6, 60, -1, 6, -.0072x^2 + .36x ) }}}