Question

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?

Answer 1

`"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)`