If the area of the region bounded by the curve y = |x-2| + |x-5| and the line y = k (where k > 3 ) is 20 sq. units, then the value of k is

Step-by-step solution

Given curve is y = f(x) = |x-2| + |x-5| . The piecewise definition of the function is: f(x) = 7 - 2x for x f(x) = 3 for 2 x 5 f(x) = 2x - 7 for x > 5 The graph of y = f(x) is a 'bucket' shape with a flat bottom from x = 2 to x = 5 at a height of y = 3 . The line y = k (with k > 3 ) intersects the curve at two points. We find the x -coordinates by equating f(x) = k : For x For x > 5 : 2x - 7 = k x = k+7 2 The bounded region is a trapezium with parallel horizontal sides. The length of the lower parallel side (on y =