A trapezium is such that three of its sides have lengths as 6 c m , then the length (in cm ) of the fourth side such that the area of trapezium is maximum, is

Step-by-step solution

From the figure above, Area of the trapezium = 6 × 6 sin ⁡ θ + 2 × 1 2 × 6 cos ⁡ θ × 6 sin ⁡ θ A = 36 sin ⁡ θ + 36 sin ⁡ θ cos ⁡ θ A = 36 sin ⁡ θ + 18 sin ⁡ 2 θ d A d θ = 0 ⇒ 36 cos ⁡ θ + 36 cos ⁡ 2 θ = 0 2 c o s 2 θ + cos ⁡ θ - 1 = 0 cos ⁡ θ = 1 2 or cos ⁡ θ = - 1 for θ = π 3 , d 2 A d θ 2 < 0 So, area is maximum when θ = π 3 Length of fourth side = 6 c o s π 3 × 2 + 6 = 12 c m