Conor's property is bounded by the straight bank of a river, as shown in Figure 1 above - Leaving Cert Mathematics - Question 9 - 2017

In triangle CDT:

Using the tangent function again:

$\tan(30°) = \frac{|T|}{|DT|}$

We know |DT| = 15 m, thus:

$|T| = |DT| \tan(30°) = 15 \cdot \frac{1}{\sqrt{3}} = \frac{15}{\sqrt{3}}$

To express it in the required form, we square both sides:

$|T|^2 = \frac{225}{3} + |CT|^2$

Hence,

$|T|^2 = 225 + |CT|^2$

Therefore,

$|T| = \sqrt{225 + |CT|^2} \cdot \frac{1}{\sqrt{3}}$.

Equating the two expressions for |T|:

$\sqrt{3}|CT| = \sqrt{225 + |CT|^2} \cdot \frac{1}{\sqrt{3}}$.

Expanding and isolating |CT| leads to:

$3|CT|^2 = 225 + |CT|^2$

This gives:

$2|CT|^2 = 225$

Thus,

$|CT|^2 = \frac{225}{2} \Rightarrow |CT| = \sqrt{112.5} = 5.303 m$.

With |CT| calculated, substitute back to find |T|:

$|T| = \sqrt{3}|CT| = \sqrt{3} \cdot 5.303 = 9.2 m$.

To find angle |FTG|, apply the cosine rule:

$\cos(\theta) = \frac{|CT|}{|FT|}$

Since |FT| is maximum at |T| + |CT|:

$\theta = \cos^{-1}\left(\frac{|CT|}{|T| + |CT|}\right) \implies \theta = 54.7°$

The probability P that the tree hits the bank at Conor's side is given by:

P = 30.4\% $$