Question

A round table cover has six equal designs as shown in Figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs 0.35 per cm\(^2\) . (Use √3 = 1.7)

Answer 1

The designs are segments of a circle.

Consider segment APB, where chord AB is a side of the hexagon.

Each chord subtends \(\frac{360°}{6}\) = 60° at the center.

In ΔOAB,

∠OAB = ∠OBA                 (Since OA = OB)

∠AOB = 60° 

∠OAB + ∠OBA + ∠AOB = 180° 

2∠OAB = 180° - 60° = 120°

∠OAB = 60° 
Thus, ΔOAB is an equilateral triangle.

Area of ΔOAB = \(\frac{\sqrt3 }{4} \times (side)^2 \)

= \(\frac{\sqrt3}{4} \times (28)^2\) = \(196 \sqrt3\) = \(196 \times 1.7 \) = 333.2 cm\(^2\)

Area of sector OAPB = \(\frac{60°}{ 360°} \times \pi r^2\)

= \(\frac{1}{6} \times \frac{22}{7} \times 28 \times 28\) = \(\frac{1232}{3}\) cm\(^2\)

Area of segment APB = Area of sector OAPB - Area of ΔOAB

∴ Area of designs = \(6 \times (\frac{1232}{3} - 333.2) \) cm\(^2\)
                              = (2464 - 1999.2) cm\(^2\)
                              = 464.8 cm\(^2\)

Cost of making 1 cm\(^2\) designs = Rs 0.35
Cost of making 464.8 cm\(^2\) designs = \(464.8 \times 0.35\) = Rs 162.68

Therefore, the cost of making these designs is Rs 162.68.