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°?
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° |
x_{1} |
x_{2} |
x_{3} |
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° = x_{1 }× 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.