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प्रश्न
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 `sqrt3 = 1.7`]
उत्तर
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 `(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 = `sqrt3/4 xx ("side")^2`
`=sqrt3/4 xx (28)^2 `
` = 196sqrt3`
` = 196 xx 1.7 `
= 333.2 cm2
Area of sector OAPB = `60^@/360^@ xx pir^2`
`= 1/6xx 22/7xx28xx28`
`= 1232/3 cm^2`
Area of segment APB = Area of sector OAPB − Area of ΔOAB
`=(1232/3 - 333.2) cm^2`
Therefore, the number of designs = `6xx(1232/3 - 333.2) cm^2`
`= (2464 - 1999.2) cm^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.
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