Question 1143669
You can put the vertex at the point ( 0, -2 ) on the y-axis.
The canal will then go through points ( 4,0 ) and ( -4,0 )
which is maximum width
The general equation is:
{{{ y =  a*x^2 + b*x + c  }}}
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( 0,-2 )
{{{ -2 = a*0^2 + b*0 + c }}}
{{{ -2 = c }}}
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( 4,0 )
{{{ 0 = a*4^2 + b*4 - 2 }}}
{{{ 0 = 16a + 4b - 2 }}}
(1) {{{ 8a + 2b = 1 }}}
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(-4,0 )
{{{ 0 = a*(-4)^2 + b*(-4) - 2 }}}
{{{ 0 = 16a - 4b - 2 }}}
(2) {{{ 8a - 2b = 1 }}}
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Add (1) and (2)
{{{ 8a + 2b = 1 }}}
{{{ 8a - 2b = 1 }}}
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{{{ 16a = 2 }}}
{{{ a = 1/8 }}}
and
(1) {{{ 8a + 2b = 1 }}}
(1) {{{ 8*(1/8) + 2b = 1 }}}
(1) {{{ 1 + 2b = 1 }}}
(1) {{{ 2b = 0 }}}
(1) {{{ b = 0 }}}
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The equation is:
{{{ y = (1/8)*x^2 - 2 }}}
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Every meter, the depths are:
(-4, 0 )
( -3, y[1] )
( -2, y[2] )
( -1, y[3] )
( 0. y[4] )
( 1, y[3] )
( 2, y[2] )
( 3, y[1] )
( 4, 0 )
----------------
( -3, y[1] )
{{{ y[1] = (1/8)*(-3)^2 - 2 }}}
{{{ y[1] = 9/8 - 16/8 }}}
{{{ y[1] = -7/8 }}}
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You can continue the process