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.

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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°?

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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, x1 = (4 x 90°)/8 = 45°,

x2 = (4 x 90°)/10 = 36°  and

x3 = (4 x 90°)/12 = 30°

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

x = (4 x90°)/15 = 24°

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°

y = (4x90)/40 = 9

Hence, the number of spokes in such a wheel is 9.