Red =  = f(x)

Blue = normal

Green = tangent



   .







Tangent line:

dy/dx =  = 



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:



    At x=7, dy/dx = 



     Now,   y = (2/3)x + b

            12 = (2/3)*7 + b   ==>  b = 22/3



     You can now write  y = (2/3)x + 22/3  for the tangent line

     Set y=0 to find the x-axis crossing:

                 0  = (2/3)x + 22/3

      Solving for x gives  x = -11   



   From this, we conclude M is at (-11,0) 

      



Normal line:

 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



       You can now write  y = (-3/2)x + b  for the normal line

      Use the fact that the normal line also passes through (7,12) to find 

      the y-intercept of the normal:

                   12 = (-3/2)(7) + b   ==>  solve for b ==>  b = 45/2

 

       We can now write   y = (-3/2)x + 45/2   for the normal line



      Now find the x-axis crossing by setting y=0:



             0 = (-3/2)x + 45/2  ==>  solve for x ==>  x = 15



               So N is at  (15, 0)



Area of PMN:

P is at (7,12)  so you can calculate  |PN| and |PM|   ... and then use the triangle area formula   A = (1/2)|PN| * |PM|



I got 156.00 sq units when I did these calculations.

   

Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!
 








At (7,12), the slope of the tangent line is , so the slope of the normal line is -3/2.


Find the x-intercepts of the tangent line and normal line by setting y=0:


















The two x-intercepts are -11 and 15, so the triangle has base length 15-(-11)=26 and height 12; its area is (1/2)(26)(12)=156


ANSWER: 156


 

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