Let S be the region bounded by the curves y = x^2 and y = x . A line y = kx (where 0 < k < 1 ) divides S into two regions. If the ratio of the area of the larger region to the area of the smaller region is 15 , then the value of 10k is equal to

Step-by-step solution

The region S is bounded by y = x^2 and y = x . The points of intersection of these curves are (0,0) and (1,1) . The total area of the region S is: Area (S) = ₀¹ ( x - x^2) dx = [ 2 3 x^ 3/2 - x^3 3 ]₀¹ = 1 3 The line y = kx intersects the curve y = x^2 when kx = x^2 , which gives x = 0 and x = k . Since 0 Let R₂ be the area of the region bounded by y = kx and y = x^2 : R₂ = ₀^ k (kx - x^2) dx = [ kx^2 2 - x^3 3 ]₀^ k = k^3 2 - k^3 3 = k^3 6 The area of the other region is R₁ = Area (S) - R₂ = 1 3 - k^3 6 . Since 0