A round table cover has six equal designs as shown in Fig. below. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per cm². (Use $\sqrt{3}$ = 1.7)

Since,

There are 6 equal chords so each chord will subtend $\dfrac{360°}{6}$ = 60° at the center.

Let AB be one of the chords and center be O.

In triangle OAB,

OM ⊥ AB

In triangle OAM and OMB,

∠OMA = ∠OMB = 90°

OA = OB (Radius of same circle)

OM = OM (Common)

∴ △OAM ≅ △OMB (By RHS axiom)

∴ ∠MOB = ∠MOA = $\dfrac{60°}{2}$ = 30°. [By C.P.C.T.]

In △MOB,

⇒ sin 30° = $\dfrac{MB}{OB}$

⇒ $\dfrac{1}{2} = \dfrac{MB}{r}$

⇒ MB = $\dfrac{r}{2} = \dfrac{28}{2}$ = 14 cm

By C.P.C.T.

MA = MB = 14 cm

AB = MA + MB = 28 cm.

Since, OA = OB = AB.

∴ △AOB is an equilateral triangle.

We know that,

Area of equilateral triangle = $\dfrac{\sqrt{3}}{4} \times$ (Side)²

$\text{Area of equilateral triangle AOB} = \dfrac{\sqrt{3}}{4} \times 28^2 \\[1em] = 1.7 \times 7 \times 28 \\[1em] = 333.2 \text{ cm}^2.$

We know that,

Area of sector of angle θ and radius r = $\dfrac{θ}{360°} \times πr^2$

$\text{Area of sector AOBP} = \dfrac{60°}{360°} \times \dfrac{22}{7} \times 28^2 \\[1em] = \dfrac{1}{6} \times 22 \times 4 \times 28 \\[1em] = \dfrac{1}{3} \times 11 \times 4 \times 28 \\[1em] = \dfrac{1232}{3} \\[1em] = 410.67 \text{ cm}^2.$

From figure,

Area of segment ABP = Area of sector AOBP - Area of triangle OAB

= 410.67 - 333.2 = 77.47 cm² ≈ 77.5 cm².

There are 6 such segment.

So, total area of segments = 77.5 × 6 = 464.82 cm².

Given,

Cost of making designs = ₹ 0.35 per cm²

So, total cost = Area × Cost per area

= 464.82 × 0.35 = ₹ 162.68

Hence, cost of making designs = ₹ 162.68