A Cow, A Monkey and a Tree

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Q: A Cow is tethered to a tree by a rope that has half the length of the circumference of the tree's stump. A monkey with a stone in hand jumps onto the tree and starts hopping around the tree's branches which covers a circular area of radius five times that of the stump. This causes the cow to run about at random and monkey drops the stone at random. What is the probability that the cow would get hit?

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A: The situation can be characterized by the figure shown below (not to scale)

The monkey can be anywhere in the light shaded orange area before it drops the stone (this of course assumes that the stump has no terrace area). The cow, can sweep a spiral as shown in the figure below along either lines (poorly drawn).

To start, lets first compute the area the cow can sweep. Assume the cow moves a bit to the left, the rope would wind up along the stump of the tree. To further formalize this assume it sweeps out an angle \(\theta\) at the center as shown figure below.

In the figure if the cow's rope winds up by a length AB subtending an angle \(\theta\) at the center C, and if the cow moves from D to E an incremental angle of \(d\theta \) is subtended at the centre. The length of the rope is now decreased from \(\pi r\) to \(\pi r - r\theta\). The length of DE can be computed as
$$
(\pi r - r\theta )d\theta
$$
The incremental area of triangle ADE is \(\frac{1}{2}\times DE \times AD\) which works out as
$$
dA = \frac{1}{2}(\pi r - r \theta)^{2}d\theta
$$
To compute the total area, we can integrate out with \(\theta\) running from \([0,\pi]\) and as the area is symmetric, we multiply by 2
$$
A =2 \times \int_{0}^{\pi}\frac{1}{2}r^{2}(\pi - \theta)^{2}d\theta
$$
The above integral simplifies to
$$
A = \frac{r^{2}\pi^{3}}{3}
$$
The total area the monkey can drop the stone is easy to compute as
$$
A_{total} = \pi (5r)^{2} - \pi r^{2} = 24\pi r^{2}
$$
The sought area and probability is simply the ratio of the two areas \(\frac{A}{A_{total}} = \frac{\pi^{3}}{24\times 3} \approx 43\%\)

Some of the best books to own to learn the art of probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
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An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
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The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
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Introduction to Probability, 2nd Edition
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Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
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