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A log round the corner

CalculusHard

A straight canal $8$ m wide meets another canal $27$ m wide at a right angle. What is the longest thin log that can be floated around the corner while staying flat on the water? Treat the log as a line segment and give its length in metres, rounded to $3$ decimal places.

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Hint

The longest log that fits equals the shortest segment that touches the inner corner and the outer walls; minimise its length over the angle.

Solution

A segment at angle $θ$ to the wall of the first canal that touches the inner corner has length $L (θ) = \frac{8}{sin θ} + \frac{27}{cos θ}$ for $0 < θ < π /2$ .
Setting $L^{'} (θ) = 0$ gives $8 cos^{3} θ = 27 sin^{3} θ$ , so $tan θ = \frac{2}{3}$ .
Then $sin θ = \frac{2}{13}$ and $cos θ = \frac{3}{13}$ , so $L = 413 + 913 = 1313$ .
Since $1313 \approx 46.8722$ , the longest log has $L \approx 46.872$ m.

Answer: $\approx 46.872$

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