A belt is fitted tightly around two wheels of radii $20 cm$ & $5 cm$ which are $5 cm$ apart. How do you show that the length of the belt is $30(pi + sqrt3) cm$ ?

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Answer 1 · 2017-11-28T13:39:47

The length of the belt is shown to be $30(pi+sqrt3)cm$ .

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Since the belt is a tangent to the radii of both circles, we can safely say:

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By isolating one of the trapeziums, we get this,

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To find $f$ ( a side of the belt not in contact with the circles ), use Pythagoras Theorem,

$f=sqrt(30^2-15^2)$
$color(white)(f)=sqrt675$
$color(white)(f)=15sqrt3$

To find $alpha$ , use cosine,

$cos(alpha)=15/30$
$alpha=cos^(-1)(15/30)$
$color(white)(alpha)=60^@$

Therefore, we get this,

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Now, let's find the $c$ ( belt in contact with large circle ),

$c=(360^@-60^@-60^@)/360^@*2*pi*20$
$color(white)(c)=2/3*2*pi*20$
$color(white)(c)=80/3pi$

Now, let's find the $d$ ( belt in contact with small circle ),

$d=(60^@+60^@)/360^@*2*pi*5$
$color(white)(d)=1/3*2*pi*5$
$color(white)(d)=10/3pi$

Finally, length of belt,

Length $=80/3pi+10/3pi+2*15sqrt3$
$color(white)(xxx..)=30pi+30sqrt3$
$color(white)(xxx..)=30(pi+sqrt3)$

Hence, the length of the belt is $30(pi+sqrt3)cm$ , shown.

P.S. It's a tad bit long, sorry! Math is quite fun and not that pointless ( get it? here's a hint: circle ) after all!

A belt is fitted tightly around two wheels of radii 20 cm20 cm & 5 cm5 cm which are 5 cm5 cm apart. How do you show that the length of the belt is 30(pi + sqrt3) cm30(pi + sqrt3) cm?