Calculate minimum width of lane A lane runs perpendicular to a road $64 ft$ wide. If it is **just possible** to carry a pole $125 ft$ long from the road into the lane, keeping it horizontal, then what should be the minimum width of the lane? I am unable to understand what properties should hold for the pole to make its way into the lane. A well-explained solution is preferred (I am a twelfth standard student).

This is a version of the moving ladder problem, though more difficult because the road and lane have different widths. Let $\alpha$ be the angle between the pole and the direction of the road. In the process of moving, $\alpha$ takes on values between $0$ and $\pi/2$.

Only $64/\sin \alpha$ of the length of the pole can fit within the road. And if the width of the lane is $x$, then $x/\cos \alpha$ of the pole will fit within the lane. Therefore, we must have $$\frac{64}{\sin\alpha} + \frac{x}{\cos\alpha}\ge 125\tag1$$ for all $\alpha$ between $0$ and $\pi/2$. Rearrange as $$x\ge 125\cos\alpha - 64\cot \alpha \tag2$$ The optimal value of $x$ (when the pole **just** fits) is the maximum of the function on the right of (2) on the interval $(0, \pi/2)$.