• # question_answer1)                 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 ${{40}^{o}}$?

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