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This is one of our many quizzes on Arc length. Click on this to access more such competitive quizzes on Circle.

Arc length Quiz 2

mathematics

 A circle having the diameter of $ 55cm$ cuts the arc of $ 55cm$. What is the central angle?

 Find the radian first and then as the options are given in radians, use the formula of arc length with angle in radians.

The arc length is given by the formula: $ s = r\theta \\ \Rightarrow \theta = \dfrac{s}{r} \\ $ Given, $ s = 55cm$ and $ r = \dfrac{d}{2} = \dfrac{{55}}{2}cm$ $ \\ \Rightarrow \theta = \dfrac{{55}}{{\dfrac{{55}}{2}}} = \dfrac{{55 \times 2}}{{55}} = 2rad$.

 The radius of a circle having an arc length of $ 4.4m$ forming a central angle of $ {72^ \circ }$ is:

 Take $ \pi = \dfrac{{22}}{7}$ and convert the given decimal value in the fraction form to simplify the calculation easily.

The arc length is given by the formula: $ \\ s = 2\pi r\left( {\dfrac{\theta }{{{{360}^ \circ }}}} \right) \\ \Rightarrow r = \dfrac{{s \times 360}}{{2\pi \theta }} \\ $ Given, $ s = 4.4m = \dfrac{{44}}{{10}}m$ and $ \theta = {72^ \circ }$. $ \\ \Rightarrow r = \dfrac{{\dfrac{{44}}{{10}} \times 360}}{{2 \times \dfrac{{22}}{7} \times 72}} = \dfrac{{44 \times 36 \times 7}}{{2 \times 22 \times 72}} = \dfrac{7}{2} = 3.5m$ 

 Consider a pendulum swinging through an angle of $ {270^ \circ }$ and making an arc of $ 66inch$ length. What is the length of the pendulum?

 The radius of the circular path of the pendulum will be equal to the length of the pendulum, so find the radius using the arc length formula.

Let us first understand the given case with the help of a diagram $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/5706a862-1353-45f2-ba28-319fd25254856553865595159382077.png"/> $ \\ $ Given, arc length for pendulum $ s = 66inch$ and $ \theta = {270^ \circ }$. $ \\ $ W.K.T $s = 2\pi r\left( {\dfrac{\theta }{{{{360}^ \circ }}}} \right) \\ \Rightarrow r = \dfrac{{s \times 360}}{{2\pi \theta }} = \dfrac{{66 \times 360 \times 7}}{{2 \times 22 \times 270}} = 14inch \\ $ 

 The circular track is darker on one side than the other. What is the distance covered by a person walking on the darker part of the track if the central angle created by the darker park is $ \dfrac{{3\pi }}{2}rad$ and the diameter of the circular path is $ 35m$? $\text{(Take } \pi = \dfrac{{22}}{7})$

 The distance covered by the person walking on the darker side is the arc length of the darker side of the circular track.

Here, we are given that $ \theta = \dfrac{{3\pi }}{2}rad$ and diameter of the track $ d = 35m$. $ \\ $ Therefore, the radius of the track will be $ r = \dfrac{d}{2} = \dfrac{{35}}{2}m$. $ \\ $ As the angle is given in radians, we will use the following formula to find the arc of length of the darker side of the track. $ \\ s = r\theta \\ \Rightarrow s = \dfrac{{35}}{2} \times \dfrac{{3\pi }}{2} = \dfrac{{35 \times 3 \times 22}}{{2 \times 2 \times 7}} = \dfrac{{161}}{2} = 80.5m \\  $ 

 Vicky bought a can of coke. He is trying to open it with the help of an opener. If the radius of the can is $ 4cm$ , what arc length should the can opener travel to open the can only half way?

 Total angle of a circle is $ {360^ \circ }$ or $ 2\pi rad$. Therefore, to open the can only half, the angle will be $ {180^ \circ }$ or $ \pi rad$.

 It is given that the radius of the can $ r = 4cm$. $ \\ $ Now, to open the half can the angle covered by the opener should be: $ \\ \theta = \dfrac{{{{360}^ \circ }}}{2} = {180^ \circ } = \pi rad \\ $  The length that should be travelled by the opener is nothing but the arc length which can be determined by the formula: $ \\ s = r\theta = 4 \times \pi = 4 \times 3.14 = 12.56cm$ 