Question

Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table. Number of spokes 4 6 8 10 12 Angle between a pair of consecutive spokes 90° 60° … … … (i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion? (ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes. (iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is 40°?

Solution

A table of the given information is as follows. Number of spokes 4 6 8 10 12 Angle between a pair of consecutive spokes 90° 60° x1 x2 x3 From the given table, we obtain 4 × 90° = 360° = 6 × 60° Thus, the number of spokes and the angle between a pair of consecutive spokes are inversely proportional to each other. Therefore, 4 × 90° = x1 × 8 Similarly, and Thus, the following table is obtained. Number of spokes 4 6 8 10 12 Angle between a pair of consecutive spokes 90° 60° 45° 36° 30° (i) Yes, the number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion. (ii)Let the angle between a pair of consecutive spokes on a wheel with 15 spokes be x. Therefore, 4 × 90° = 15 × x Hence, the angle between a pair of consecutive spokes of a wheel, which has 15 spokes in it, is 24°. (iii) Let the number of spokes in a wheel, which has 40º angles between a pair of consecutive spokes, be y. Therefore, 4 × 90° = y × 40° Hence, the number of spokes in such a wheel is 9. MathematicsMathStandard VIII

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