document.write( "Question 1194922: the tangent and normal to the curve y=4*sqrt(x+2) at the point P(7,12) cut the x-axis at M and N respectively. Calculate the area of the triangle PMN \n" ); document.write( "

Algebra.Com's Answer #827222 by math_helper(2461) $\"\"$ $\"About$

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document.write( "Red =  = f(x)
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document.write( "Blue = normal
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document.write( "Green = tangent\r
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document.write( "   .\r
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document.write( "Tangent line:
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document.write( "dy/dx =  = \r
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document.write( "This is the slope of the tangent line.  Using the fact that the tangent meets the function f(x) at (7,12) let's us find the y-intercept:\r
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document.write( "    At x=7, dy/dx = \r
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document.write( "     Now,   y = (2/3)x + b
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document.write( "            12 = (2/3)*7 + b   ==>  b = 22/3\r
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document.write( "     You can now write  y = (2/3)x + 22/3  for the tangent line
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document.write( "     Set y=0 to find the x-axis crossing:
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document.write( "                 0  = (2/3)x + 22/3
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document.write( "      Solving for x gives  x = -11   \r
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document.write( "   From this, we conclude M is at (-11,0) 
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document.write( "      \r
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document.write( "Normal line:
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document.write( " The slope of the normal line will be  -1/(slope of tangent).  We are only interested in (7,12), so slope of normal is  -(1/(2/3)) = -3/2\r
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document.write( "       You can now write  y = (-3/2)x + b  for the normal line
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document.write( "      Use the fact that the normal line also passes through (7,12) to find 
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document.write( "      the y-intercept of the normal:
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document.write( "                   12 = (-3/2)(7) + b   ==>  solve for b ==>  b = 45/2
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document.write( "       We can now write   y = (-3/2)x + 45/2   for the normal line\r
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document.write( "      Now find the x-axis crossing by setting y=0:\r
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document.write( "             0 = (-3/2)x + 45/2  ==>  solve for x ==>  x = 15\r
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document.write( "               So N is at  (15, 0)\r
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document.write( "Area of PMN:
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document.write( "P is at (7,12)  so you can calculate  |PN| and |PM|   ... and then use the triangle area formula   A = (1/2)|PN| * |PM|\r
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document.write( "I got 156.00 sq units when I did these calculations.
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