As shown in the figure, in RT △ ABC, ∠ ACB = 90 °, ab = 10, BC = 8, point d moves on BC (does not move to B, c), de ‖ AC, intersects AB with E, let BD = x, the area of △ ade is y. (1) find the functional relationship between Y and X and the value range of independent variable x; (2) when x is the value, the area of △ ade is the largest? What is the maximum area?

As shown in the figure, in RT △ ABC, ∠ ACB = 90 °, ab = 10, BC = 8, point d moves on BC (does not move to B, c), de ‖ AC, intersects AB with E, let BD = x, the area of △ ade is y. (1) find the functional relationship between Y and X and the value range of independent variable x; (2) when x is the value, the area of △ ade is the largest? What is the maximum area?

(1) In RT △ ABC, AC = AB2 − BC2 = 102 − 82 = 6, ∵ tanb = 68 = 34. ∵ de ∥ AC, ∵ BDE = ∠ BCA = 90 °. ∵ de = BD · tanb = 34x, CD = bc-bd = 8-x. let the height of de edge in △ ade be h, ∵ de ∥ AC, ∥ H = CD. ∵ y = 12de · CD = 12 × 34x · (8-x), that is y = − 38x2 + 3x. The range of independent variable x is 0 ＜ x ＜ 8; (2) when x = − 32 × (− 38) = 4, y max = 4 × (− 38) × 0 When x = 4, the maximum area of △ ade is 6