question_answer 13)

                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 of pair of consecutive spokes	\[{{90}^{\text{o}}}\]	\[{{60}^{\text{o}}}\]	?.	?.	?.

(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 \[{{40}^{o}}\]?

Answer:

        \[4\times {{90}^{\text{o}}}=8\times x\] \[\Rightarrow \]               \[x=\frac{4\times {{90}^{\text{o}}}}{8}={{45}^{\text{o}}}\] \[4\times {{90}^{\text{o}}}=10\times y\] \[\Rightarrow \]               \[y=\frac{4\times {{90}^{\text{o}}}}{10}\,={{36}^{\text{o}}}\]     \[4\times {{90}^{o}}=12\times z\] \[\Rightarrow \]               \[z=\frac{4\times {{90}^{\text{o}}}}{12}={{30}^{\text{o}}}\]

Number of spokes	4	6	8	10	12
Angle between of pair of consecutive spokes	\[{{90}^{\text{o}}}\]	\[{{60}^{\text{o}}}\]	\[{{45}^{\text{o}}}\]	\[{{36}^{\text{o}}}\]	\[{{30}^{\text{o}}}\]

(i) Yes ! The number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion. \[[\because \,4\times {{90}^{\text{o}}}=6\times {{60}^{\text{o}}}=8\times {{45}^{\text{o}}}=10\times {{36}^{\text{o}}}=12\times {{30}^{\text{o}}}]\](ii) Let the angle between a pair of consecutive spokes on a wheel with 15 spokes be \[{{x}^{o}}\]. Lesser the number of spokes, more will be the angle between a pair of consecutive spokes. So, this is a case of inverse proportion.                 Hence, \[4\times {{90}^{\text{o}}}=15\times x\,[{{x}_{1}},\,{{y}_{1}}={{x}_{2}}{{y}_{2}}]\] \[\Rightarrow \]               \[x=\frac{4\times {{90}^{\text{o}}}}{15}\] \[\Rightarrow \]               \[x={{24}^{\text{o}}}\] Hence, the angle between a pair of consecutive spokes on a wheel with 15 spokes is \[{{24}^{o}}\]. (iii) Let \[x\] spokes be needed. Lesser the number of spokes, more will be the angle between a pair of consecutive spokes. So, this is a case of inverse proportion. Hence, \[4\times {{90}^{\text{o}}}=x\times {{40}^{\text{o}\,}}[{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}]\] \[\Rightarrow \]               \[x=\frac{4\times {{90}^{\text{o}}}}{{{40}^{\text{o}}}}\] \[\Rightarrow \]               \[x=9\] Hence, 9 spokes would be needed.