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°?

Answer 1

 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, 
\(4 × 90° = x_1 × 8\)
\(x_1 = \frac{4 × 90°}{8} = 45°\)

Similarly,
\(x_2 =\frac{ 4 × 90°}{10} = 36°\)
and 
\(x_3 = \frac{4 × 90°}{12} = 30°\)
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,
\(4 × 90° = 15 × x\)
\(x =\frac{ 4 × 90°}{15} = 24°\)

The angle between the pair of consecutive spokes on a wheel with 15 spokes is 24°.

---

(iii) Let the number of spokes in a wheel, which has 40 º angles between a pair of consecutive spokes, be y. 
Therefore,
\(4 × 90° = y × 40°\)
\(y =\frac{ 4 × 90°}{40} = 9\)
If the angle between a pair of consecutive spokes is 40°, then the spokes on the wheel are 9.