Two wheels of radii 2 and 6 are mutually tangent. A belt is wrapped tightly around them. What is the total length of the belt?

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1 Answer
Jan 4, 2018

#"Total length"=8(pi+sqrt5)#

Explanation:

We know that the belt will cover half the circumference of both circles, and the circumference is given by #c=rtheta#. For half a circle, the angle is #180^circ-=picolor(white)(l)"radians"#

So, we have #c_1=6pi# and #c_2=2pi#

Now, we have the section between the circles this can be found by using Pythagorus. Since we can create a right angled triangle which has a base of the two radii, #2# and #6#, and a height equal to the difference of the two radii, #6-2=4#.

So, we have #a^2=b^2+c^2#, and #a=sqrt(8^2+4^2)=sqrt(80)#.

However, we have two of these, so the whole distance would be #d=c_1+c_2+2a=6pi+2pi+2sqrt(80)=8pi+2sqrt(80)=8pi+2(4sqrt(5))=8pi+8sqrt(5)=8(pi+sqrt5)#