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 cm². [Use ${\displaystyle \sqrt{3}=1.7}$]

Answer 1

It can be observed that these designs are segments of the circle.

Consider segment APB. Chord AB is a side of the hexagon. Each chord will substitute ${\displaystyle \frac{360º}{6}}$ = 60º at the centre of the circle.

In ΔOAB,

∠OAB = ∠OBA (As OA = OB)

∠AOB = 60°

∠OAB + ∠OBA + ∠AOB = 180°

2∠OAB = 180° − 60° = 120°

∠OAB = 60°

Therefore, ΔOAB is an equilateral triangle.

Area of ΔOAB = ${\displaystyle \frac{\sqrt{3}}{4}\times {\left(\text{side}\right)}^2}$

${\displaystyle =\frac{\sqrt{3}}{4}\times {\left(28\right)}^2}$

${\displaystyle =196\sqrt{3}}$

${\displaystyle =196\times 1.7}$

= 333.2 cm²

Area of sector OAPB = ${\displaystyle \frac{{60}^{\circ }}{{360}^{\circ }}\times \pi r^2}$

${\displaystyle =\frac{1}{6}\times \frac{22}{7}\times 28\times 28}$

${\displaystyle =\frac{1232}{3}c{m}^2}$

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

${\displaystyle =(\frac{1232}{3}-333.2)c{m}^2}$

Therefore, the number of designs = ${\displaystyle 6\times (\frac{1232}{3}-333.2)c{m}^2}$

${\displaystyle =(2464-1999.2)c{m}^2}$

= 464.8 cm²

Cost of making 1 cm² designs = Rs 0.35

Cost of making 464.76 cm² designs = 464.8 × 0.35 = Rs 162.68

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