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Question Number 43531 by pieroo last updated on 11/Sep/18

the circumference of a circular track is 9 km. A cyclist rides round it a number of times and stops after covering a distance of 302 km. How far is the cyclist from the starting point? [take Π=((22)/7)]

$$\mathrm{the}\:\mathrm{circumference}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circular}\:\mathrm{track}\:\mathrm{is}\:\mathrm{9}\:\mathrm{km}.\:\mathrm{A}\:\mathrm{cyclist} \\ $$$$\mathrm{rides}\:\mathrm{round}\:\mathrm{it}\:\mathrm{a}\:\mathrm{number}\:\mathrm{of}\:\mathrm{times}\:\mathrm{and}\:\mathrm{stops}\:\mathrm{after} \\ $$$$\mathrm{covering}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{302}\:\mathrm{km}.\:\mathrm{How}\:\mathrm{far}\:\mathrm{is}\:\mathrm{the}\:\mathrm{cyclist} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{starting}\:\mathrm{point}?\:\left[\mathrm{take}\:\Pi=\frac{\mathrm{22}}{\mathrm{7}}\right] \\ $$

Commented by pieroo last updated on 12/Sep/18

I can hardly interprete the solution. But thanks anyway

$$\mathrm{I}\:\mathrm{can}\:\mathrm{hardly}\:\mathrm{interprete}\:\mathrm{the}\:\mathrm{solution}.\:\mathrm{But}\:\mathrm{thanks}\:\mathrm{anyway} \\ $$

Answered by MrW3 last updated on 11/Sep/18

302 mod 9 = 5 km or −4 km 2πR=9 ⇒R=(9/(2×((22)/7)))=((63)/(44)) km θR=4 ⇒θ=(4/R)=((176)/(63)) rad=160° L=2Rsin (θ/2)=2×((63)/(44))×sin 80°=2.82 km ⇒the cyclist is 2.82 km far from starting point.

$$\mathrm{302}\:{mod}\:\mathrm{9}\:=\:\mathrm{5}\:{km}\:{or}\:−\mathrm{4}\:{km} \\ $$$$\mathrm{2}\pi{R}=\mathrm{9} \\ $$$$\Rightarrow{R}=\frac{\mathrm{9}}{\mathrm{2}×\frac{\mathrm{22}}{\mathrm{7}}}=\frac{\mathrm{63}}{\mathrm{44}}\:{km} \\ $$$$\theta{R}=\mathrm{4} \\ $$$$\Rightarrow\theta=\frac{\mathrm{4}}{{R}}=\frac{\mathrm{176}}{\mathrm{63}}\:{rad}=\mathrm{160}° \\ $$$${L}=\mathrm{2}{R}\mathrm{sin}\:\frac{\theta}{\mathrm{2}}=\mathrm{2}×\frac{\mathrm{63}}{\mathrm{44}}×\mathrm{sin}\:\mathrm{80}°=\mathrm{2}.\mathrm{82}\:{km} \\ $$$$\Rightarrow{the}\:{cyclist}\:{is}\:\mathrm{2}.\mathrm{82}\:{km}\:{far}\:{from} \\ $$$${starting}\:{point}. \\ $$

Commented by MrW3 last updated on 12/Sep/18

Commented by MrW3 last updated on 12/Sep/18

A=starting point B=finish point L=AB^(__) =distance btw. A and B with AB^(⌢) =4 km

$${A}={starting}\:{point} \\ $$$${B}={finish}\:{point} \\ $$$${L}=\overset{\_\_} {{AB}}={distance}\:{btw}.\:{A}\:{and}\:{B} \\ $$$${with}\:\overset{\frown} {{AB}}=\mathrm{4}\:{km} \\ $$

Commented by pieroo last updated on 12/Sep/18

yea its clear now. thanks very much

$$\mathrm{yea}\:\mathrm{its}\:\mathrm{clear}\:\mathrm{now}.\:\mathrm{thanks}\:\mathrm{very}\:\mathrm{much} \\ $$

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