| Number of spokes | 4 | 6 | 8 | 10 | 12 |
| Angle between a pair of consecutive spokes | \[90{}^\circ \] | \[60{}^\circ \] | - | - | - |
| (a) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion? | |||||
| (b) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes. | |||||
| (c) How many spokes would be needed, if the angle between a pair of consecutive spokes is\[40{}^\circ \]? | |||||
Answer:
(a) Suppose number of spokes be x and angle between a pair of consecutive spokes be y.
X 4 6 8 10 12 y 90 60 \[{{y}_{1}}\] \[{{y}_{2}}\] \[{{y}_{3}}\] As the number of spokes increase, angle between a pair of consecutive spokes decreases. Hence, it is a case of inverse proportion. i.e., \[{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}\] (i) \[{{x}_{1}}=\text{ }6,{{x}_{2}}=\text{ }8,{{y}_{1}}=\text{ }60,{{y}_{2}}=\text{ }?\] \[6\times 60{}^\circ =\text{ }8{{y}_{1}}\] \[\Rightarrow \] \[{{y}_{1}}=\frac{360{}^\circ }{8}=45{}^\circ \] (ii) \[8\times 45{}^\circ =10\times {{y}_{2}}\] \[\Rightarrow \] \[{{y}_{2}}=\frac{360{}^\circ }{10}=36{}^\circ \] (iii) \[10\times 36{}^\circ =12\times {{y}_{3}}\] \[\Rightarrow \] \[{{y}_{3}}=\frac{360{}^\circ }{12}=30{}^\circ \] The table is
X 4 6 8 10 12 y \[90{}^\circ \] \[60{}^\circ \] \[45{}^\circ \] \[36{}^\circ \] \[30{}^\circ \] The number of spokes and the angles formed between the pairs of consecutive spokes is in inverse proportion. Because the products of the corresponding values of two quantities is constant. i.e\[~4\times 90{}^\circ =6\times 60{}^\circ =8\times 45{}^\circ =10\times 36{}^\circ =12\times 30{}^\circ =360{}^\circ \]
(b) Let the angle be x . The following table
No. of spokes 12 15 Angle \[30{}^\circ \] x As number of spokes increases, the angle decreases it is the case of inverse proportion. i.e. \[{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}\] \[12\times 30{}^\circ =15\times x\] \[\Rightarrow \] \[x=\frac{12\times 30{}^\circ }{15}=24{}^\circ \] Thus, angle is\[24{}^\circ .\]
(c) Let the spokes be x, if the angle between a pair of consecutive spokes is \[40{}^\circ .\] We have the following table.
No. of spokes 4 6 8 X 10 12 Angle \[90{}^\circ \] \[60{}^\circ \] \[45{}^\circ \] \[40{}^\circ \] \[36{}^\circ \] \[30{}^\circ \] As no. of spokes increases the angle decreases. Hence, it is the case of inverse proportion We have, \[8\times 45{}^\circ =x\times \text{ }40{}^\circ \] \[\Rightarrow \] \[x=\frac{8\times 45}{40{}^\circ }\] \[\Rightarrow \] x=9 Hence, the number of spokes are 9.
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