Question 1155851
Let {{{ d }}} = the diameter of the base
Let {{{ h }}} = the height of the tank
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The circumference, {{{ C }}}, of the base is:
{{{ C = pi*d }}}
The area, {{{ A[b] }}}, of the base is:
{{{ A[b] = pi*(d/2)^2 }}}
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The volume, {{{ V }}} is:
{{{ V = pi*( d/2 )^2 * h }}}
{{{ 16000*pi = pi*( d^2/4 )*h }}}
{{{ 64000 = d^2*h }}}
{{{ h = 64000 / d^2 }}}
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The area of the sides, {{{ A[s] }}}, is:
{{{ A[s] = pi*d*( 64000 / d^2 ) }}}
{{{ A[s] = ( 64000*pi ) / d }}}
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The cost, {{{ C[t] }}}, of the tank is:
{{{ C[t] = 8.64*(( 64000*pi )/d ) + 10*( pi*d^2 ) / 4 }}}
{{{ C[t] = 1737175.07 / d + 7.85398*d^2 }}}
The slope of this curve is:
{{{ C[tt] = -1737175.07 / d^2 + 15.70796d }}}
Set slope = {{{0}}}
{{{ 0 = -1737175.07 / d^2 + 15.70796d }}}
{{{ 1737175.07 / d^2 =  15.70796d }}}
{{{ d^3 =  1737175.07 / 15.70796 }}}
{{{ d = 120.21201 / 2.5044167 }}}
{{{ d = 48.0000003 }}}
{{{ d = 48 }}} ft
and
{{{ h = 64000 / d^2
{{{ h = 64000 / 2304 }}}
{{{ h = 27.778 }}} ft
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The diameter for min cost is 48 ft
The height for min cost is 27.778 ft
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Here's the plot:
{{{ graph( 500, 500, -250, 250, -100000, 350000, 1737175.07/x + 7.85398x^2 ) }}}
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The min cost is:
{{{ C[t] = 1737175.07 / d + 7.85398*d^2 }}}
{{{ C[t] = 1737175.07 / 48 + 7.85398*48^2 }}}
{{{ C[t] = 36191.15 + 18095.57 }}}
{{{ C[t] = 54286.72 }}}
$54,286.72 looks close to min in the plot
Get a 2nd opinion if needed
and check my math