#### 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° | ... | ... | ... |

1) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion?

2) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes.

3) 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° | 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

`x_1 = (4 xx 90^@)/8 = 45^@`

Similary, `x_2 = (4 xx 90^@)/10 = 36^@` and `x_3 = (4 xx90^@)/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° | 40° | 36° | 30° |

1) Yes, the number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion.

2) Let the angle between a pair of consecutive spokes on a wheel with 15 spokes be *x*. Therefore,

4 × 90° = 15 × *x*

x = `(4 xx 90^@)/15 = 24^@`

Hence, the angle between a pair of consecutive spokes of a wheel, which has 15 spokes in it, is 24°.

3) 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 = `(4 xx 90)/40 = 9`

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