Question 1085512
Let {{{ x }}} = the length of a partition
{{{ 4x }}} = the total length of the 2 partitions
plus the ends that are parallel to the 2 partitions
{{{ y }}} = the length of the other side of the pen
{{{ 2y + 4x = 1000 }}} 
{{{ 2y = -4x + 1000 }}}
{{{ y = -2x + 500 }}}
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(a)
{{{ A = x*y }}}
{{{ A = x*( -2x + 500 ) }}}
{{{ A = -2x^2 + 500x }}}
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(b)
This is a parabola with a maximum. Use the formula:
{{{ x[max] = -b/(2a) }}}
{{{ a = -2 }}}
{{{ b = 500 }}}
{{{ x[max] = -500/( 2*(-2)) }}}
{{{ x[max] = 125 }}}
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Now find {{{ y[max] }}}
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{{{ y[max] = -2*125 + 500 }}}
{{{ y[max] = 250 }}}
The dimensions for maximum area are:
125 x 250
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(c)
{{{ 125*250 = 31250 }}}
31,250 ft2
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check the answer:
{{{ 2y + 4x = 1000 }}} 
{{{ 2*250 + 4*125 = 1000 }}}
{{{ 500 + 500 = 1000 }}}
{{{ 1000 = 1000 }}}
OK