In the figure (i) given below, the boundary of the shaded region in the given diagram consists of four semicircular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, calculate

(i) the length of the boundary.

(ii) the area of the shaded region.

(i) From figure,

OA = $\dfrac{AD}{2} = \dfrac{14}{2}$ = 7 cm.

OB = OA - AB = 7 - 3.5 = 3.5 cm.

Radius of smallest semi-circle = $\dfrac{3.5}{2}$ = 1.75 cm.

Circumference of semi-circle = πr.

Length of boundary = Circumference of largest semi-circle + Circumference of smaller semi-circle + 2 × Circumference of smallest semi-circle

= 7π + 3.5π + (2 x 1.75π)

= 7π + 3.5π + 3.5π

= 14π

= $14 \times \dfrac{22}{7}$

= 2 × 22

= 44 cm.

Hence, length of boundary = 44 cm.

(ii) Area of shaded region = Area of large semi-circle + Area of smaller semi-circle - 2 × Area of smallest semi-circle

$= \dfrac{π(7)^2}{2} + \dfrac{π(3.5)^2}{2} - 2 \times \dfrac{π(1.75)^2}{2} \\[1em] = \dfrac{49π}{2} + \dfrac{12.25π}{2} - \dfrac{6.125π}{2} \\[1em] = \dfrac{49π + 12.25π - 6.125π}{2} \\[1em] = \dfrac{55.125π}{2} \\[1em] = \dfrac{55.125 \times \dfrac{22}{7}}{2} \\[1em] = \dfrac{1212.75}{14} \\[1em] = 86.625 \text{ cm}^2.$

Hence, area of shaded region = 86.625 cm².