Ex.13.2 Q3 Direct and Inverse Proportions Solution - NCERT Maths Class 8

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Question

Rehman is making a wheel using spokes. He wants to fix equal spokes n such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table.

(i) Are the number of spokes and the angle 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 \(40^\circ?\)

Number of spokes \(4 \) \(6\) \(8\) \(10\) \(12\)

Angle between a pair

of consecutive  spokes 

 \(90^\circ\) \(60^\circ\) \(\dots\) \(\dots\) \(\dots\)

Text Solution

What is known?

(i) Number of spokes \(4\) and \(6.\)

(ii) Angle between the spokes is \(90⁰\) and \(60⁰.\)

What is unknown?

Angle between the spokes for spoke \(8, 10\) and \(12.\)

Reasoning:

Two numbers \(x\) and \(y\) are said to vary in inverse proportion if,

\[\begin{align}x y = {\rm{ }}k,{\rm{ }}\,\,x{\rm{ }} = {\rm{ }}\frac{1}{y}k\end{align}\]

Where \(k\) is a constant.

\[{x_1}{y_1} = {x_2}{y_2}\]

Steps:

(i) If the number of spoke increases, then the angle between the spoke decreases. Hence it is in inverse variation.

(ii)

\[\begin{align}{x_1}{y_1} &= {x_3}{y_3}\\4 \times {90^{\rm{o}}} &= 8 \times {y_3}\\{y_3} &= \frac{{4 \times {{90}^{\rm{o}}}}}{8} = {45^{\rm{o}}}\\{x_1}{y_1} &= {x_4}{y_4}\\4 \times {90^{\rm{o}}} &= 10 \times {y_4}\\{y_4} &= \frac{{4 \times 90^\circ }}{{10}} = {36^{\rm{o}}}\\{x_1}{y_1} &= {x_5}{y_5}\\4 \times {90^{\rm{o}}} &= 12 \times {y_5}\\{y_5} &= \frac{{4 \times {{90}^{\rm{o}}}}}{{12}} = {30^{\rm{o}}}\end{align}\]

Number of spokes

\(8\)

\(10\)

\(12\)

Angle between spokes

\(30^\circ\)

\(36^\circ\)

\(45^\circ\)

(i) Are the number of spokes and the angle formed between the pairs of consecutive spokes in inverse proportion?

Ans: Yes, number of spokes and the angle formed between the pairs of consecutive spokes in inverse proportion, because for \(4\) spokes the angle is \(40\) but for \(8\) spokes it is \(45.\)

(ii) Calculate the angle between a pair of consecutive spokes on a wheel with \(15\) spokes. More the number of spokes,less  the angle between them.

\[\begin{align}{x_1}{y_1} &= {x_2}{y_2}\\4 \times {90^{\rm{o}}} &= 15 \times {y_2}\\{y_2}& = \frac{{4 \times {{90}^{\rm{o}}}}}{{15}}\\ &= {24^{\rm{o}}}\end{align}\]

The angle between the pair of consecutive spokes on a wheel with \(15\) spokes is \({24^{\rm{o}}}.\)

(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is \({40^{\rm{o}}}.\)

\[\begin{align}{x_1}{y_1} &= {x_2}{y_2}\\4{\rm{ }} \times {\rm{ }}{90^o} &= {\rm{ }}{x_2} \times 40\\{x_2} &= \frac{{4 \times 90^\circ }}{{40}}\\ &= 9\end{align}\]

If the angle between a pair of consecutive spokes is \({40^{\rm{o}}},\) then spokes on the wheel is \(9.\)