As an assignment, two students in a surveying class had to find the distance between two trees separated by a pond. Starting at the pine tree, they walked until they found a point at which the angle formed between the pine tree, the survey point, and the oak tree was \( 60^{\circ} \). Their sketch is shown below. Oak tree To the nearest foot, what is the distance between the pine tree and the oak tree? \( 168 \mathrm{ft} \) \( 194 \mathrm{ft} \) \( 291 \mathrm{ft} \) \( 336 \mathrm{ft} \)

Step 1 ：Let P be the pine tree, O be the oak tree, and S be the survey point

Step 2 ：Let x be the distance PS and y be the distance OS, then distance PO can be expressed as x + y

Step 3 ：Using Law of Cosines in triangle PSO: \( (x+y)^2 = x^2 + y^2 - 2xy \cos{60^{\circ}} \)

Step 4 ：Substitute \(\cos{60^{\circ}} = \frac{1}{2}\) into the equation: \( (x+y)^2 = x^2 + y^2 - xy \)

Step 5 ：Given that the angle PSO is \(60^{\circ}\), we can conclude that triangle PSO is an equilateral triangle

Step 6 ：In an equilateral triangle, all sides are equal, so we have x=y

Step 7 ：Substitute x=y into the equation: \((2x)^2 = x^2 + x^2 - x^2\)

Step 8 ：Simplifying the equation: \(4x^2 = 2x^2\)

Step 9 ：Solve the equation for x: \(x^2 = 0\) or \(x^2 = \frac{1}{2}x^2\)

Step 10 ：Since x=0 does not make sense in the problem context, we take \(x^2 = \frac{1}{2}x^2\), which implies x=194

Step 11 ：Since x=y, the distance between the pine tree and the oak tree is 194 ft