Trigonometry problem involving cylinders.

[coderot]

Homework Statement

Three cylinders are placed in contact with one another with their axes parallel.
The radii of the cylinders are 3, 4 and 5 cm. An elastic band is stretched around
the three cylinders so that the plane of the band is perpendicular to the axes of
the cylinder. Calculate the length of band that is in contact with the largest
cylinder.

Homework Equations

Sine and Cosine Rules. Properties of circles i.e. arc length and area sector etc. I've also attached an image of what I think this problem looks like.

The Attempt at a Solution

I started this by solving the triangle using the cosine and sine rules. This gave me the following angles: A = 48.2, B = 58.4 and C = 73.4. Note: answers given in degrees.

Now I don't know how to proceed.

 

[LCKurtz]

Homework Statement

Three cylinders are placed in contact with one another with their axes parallel.
The radii of the cylinders are 3, 4 and 5 cm. An elastic band is stretched around
the three cylinders so that the plane of the band is perpendicular to the axes of
the cylinder. Calculate the length of band that is in contact with the largest
cylinder.

Homework Equations

Sine and Cosine Rules. Properties of circles i.e. arc length and area sector etc. I've also attached an image of what I think this problem looks like.

The Attempt at a Solution

I started this by solving the triangle using the cosine and sine rules. This gave me the following angles: A = 48.2, B = 58.4 and C = 73.4. Note: answers given in degrees.

Now I don't know how to proceed.

You don't need to know any trig functions. Remember the radius is perpendicular to its tangent line at the point of tangency. If you draw a line between centers of two of those circles and the two radii to the tangent between them you get a trapezoid with 3 sides known and two right angles. The tangent is the 4th side and you can get its length with the pythagorean theorem.

[Edit] Reading too quickly I thought you wanted the part not in contact with the cylinders. More later...

[Edit2] If you draw a parallel to the tangent line but passing through the nearest center, you can read the other angles you need off the triangle it forms at the top of the trapezoid.

 

[coderot]

Okay,

Used your second edit I produced the diagram attached. This got me the correct solution. Thankyou. :)

However I had to use trigonometry to get there. I essentially worked out the other two angles (using trig) around A, added these to angle A, and then subtracted this from 360. It was then a case of finding the arc length. You mentioned that I wouldn't have needed any trig. Would you care to explain how you could have got the same result as me? (I got 13.32cm btw).

Thanks again.

 

[LCKurtz]

Okay,

Used your second edit I produced the diagram attached. This got me the correct solution. Thankyou. :)

However I had to use trigonometry to get there. I essentially worked out the other two angles (using trig) around A, added these to angle A, and then subtracted this from 360. It was then a case of finding the arc length. You mentioned that I wouldn't have needed any trig. Would you care to explain how you could have got the same result as me? (I got 13.32cm btw).

That was when I thought the problem was to get the length that wasn't touching the cylinders. You just didn't need trig for that part. What you have done looks like what I meant in the second hint.