document.write( "Question 9590: A right triangle has one vertex on the graph of y = 3 - x^2, x > 0, and (x, y), another at the origin, and the third on the positive x-axis at (x, 0). Express the area A of the triangle as a function of x. \n" ); document.write( "
Algebra.Com's Answer #5217 by prince_abubu(198)\"\" \"About 
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\"+graph%28200%2C200%2C-4%2C4%2C-4%2C4%2C3-x%5E2%29+\"\r
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\n" ); document.write( "\n" ); document.write( "The graph above is for \"+y+=+3+-+x%5E2+\". One of your vertex lies on the curve. I apologize for the fact that it's quite difficult to draw a triangle that meets what the problem is saying. However, I'll do my best to explain it in words.\r
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\n" ); document.write( "\n" ); document.write( "You know that the base of the triangle will ALWAYS lie in the x-axis. Its length (the base) is ALWAYS the x, since you're measuring from the origin to wherever x is.\r
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\n" ); document.write( "\n" ); document.write( "Now the height of your triangle will depend on where your x is, and what the function \"+y+=+3+-+x%5E2+\" will do to it. Actually, the height of your triangle will be the \"+3+-+x%5E2+\". Now, hang on because this is where it'll get tricky. After \"+x+=+sqrt%283%29+\", \"+3+-+x%5E2+\" will have a negative value. The only thing that happens is that your triangle becomes upside-down, and it'll still have an area. We will force the \"+3+-+x%5E2+\" to be positive by putting it inside absolute value: \"+abs%283+-+x%5E2%29+\". This way, if \"+x+%3E+sqrt%283%29+\", you'll have a positive height for your upside-down triangle.\r
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\n" ); document.write( "\n" ); document.write( "So, the area of the triangle is \"+A%5BT%5D+=+%281%2F2%29B%2AH+\". The base B is the x. The height is \"+abs%283-x%5E2%29+\". Since the area depends on x and the function \"+y+=+3+-+x%5E2+\", the area of the triange becomes a function of x:\r
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\n" ); document.write( "\n" ); document.write( "\"+A%5BT%5D%28x%29+=+%281%2F2%29x%2Aabs%283+-+x%5E2%29+\"
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