document.write( "Question 1083342: the area enclosed by the curve y=ax(1-x) (a>zero) and x-axis is divided into two equal parts by the curve y=x^2
\n" ); document.write( "find the value of a \n" ); document.write( "

Algebra.Com's Answer #697580 by htmentor(1343) $\"\"$ $\"About$

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The function y = ax - ax^2 intersects the x-axis at the points 0 and 1, regardless of the value of a.
\n" ); document.write( "The area enclosed by this curve and the x-axis is given by: $\"int%28%28ax-ax%5E2%29%2C+dx%2C+0%2C1%29%29\"$ . Let us call this area A1.
\n" ); document.write( "The curve y = x^2 divides this area into two parts
\n" ); document.write( "The area of the second part is the sum of $\"int%28+x%5E2%2C+dx%2C+0%2C+c+%29\"$ and $\"int%28%28ax-ax%5E2%29%2C+dx%2C+c%2C1%29%29\"$ ,
\n" ); document.write( "where c is the intersection point of the two curves. Let's call this area A2.
\n" ); document.write( "So we need to find the value of a for which A1 - A2 = A2, or A1 = 2A2.
\n" ); document.write( "The intersection point, c, of the two curves is ax - ax^2 = x^2 -> x = a/(a+1)
\n" ); document.write( "Performing all the integrations and simplifying, you should be left with the following equation:
\n" ); document.write( "a^2 - 2a - 1 = 0
\n" ); document.write( "This has solutions a = -0.4142 and a = 2.4142. Since a>0, we take the positive solution, a = 2.4142
\n" ); document.write( "The exact value for a is a = 1 + sqrt(2) (Check for yourself)\r
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