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° | 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.