Question 700118
Let {{{ x }}} = the length in feet of the
side parallel to the house.
The length of each of the other 2 sides is {{{ (150 - x ) / 2 }}}
The area is:
{{{ A = x*( 150 - x ) / 2 }}}
{{{ A = 75x - (1/2)*x^2 }}}
The general form is:
{{{ A = b*x + a*x^2 }}} where
{{{ a = -1/2 }}}
{{{ b = 75 }}}
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The x-co-ordinate of the vertex is at
{{{ -b/(2a) }}}
{{{ -b/(2a) = -75 / ( 2*(-1/2)) }}}
{{{ -b/(2a) = 75 / 1 }}}
{{{ x = 75 }}}
{{{ ( 150 - x ) / 2 = ( 150 - 75 ) / 2 }}}
{{{ ( 150 - x ) / 2 = 75/2 }}}
So, the 3 sides are {{{ 75 }}}, {{{ 75/2 }}}, and {{{ 75/2 }}}
The maximum area is:
{{{ A = 75*( 75/2 ) }}}
{{{ A = 2812.5 }}} ft2
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Here's the plot of the area equation:
{{{ graph( 400, 400, -20, 160, -300, 3200, -(1/2)*x^2 + 75x ) }}}