calculating areas that cow can graze

[gmencl]

I have 3 questions dealing with areas that I need help with.

1. Clara the cow has been tied to a circular silo with radius r by a rope just long enough to reach a point diametrically opposite to the point where she is tied. If she goes to the left side of the silo, she can stand far away from the silo, while at the right side, she can only graze right next to the silo. We wish to compute the total area of the region upon which she can graze.

1. How far from the silo can the cow stand when she is to its left?

Assume that a very long rope is wound around the circular silo, and then unwound while being held taut. The curve traced by the end of the rope is called the involute of the circle. if the silo has radius r and center O, and if the parameter theta is chosen,
2. show that the parametric equations of the involute are:
x = r(cos(theta) + theta sin(theta)
y = r(sin(theta) - theta cos(theta)

3. What is the area of the grazing region available to the cow?
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[galactus]

I believe, if I understand the problem correctly, you'll have an area similar to a cardioid with a circle of radius r in side. Like so:

This is just a representation. The inside circle is the silo.

If you tie the cow at a point on the circle at x=2, then draw a tangent line at that point, the cow can eat a semicircle of radius, say, L.
But, the cow can eat around the silo where the tether wraps around the silo.

So you have three regions to find the area of.

The area of the semicircle is easy. It has area \(\displaystyle \frac{{\pi}L^{2}}{2}\)

Can you find the other two areas when the cow wraps around the silo?.