MathProblem 32 A fence, board, and a shadow problem

You have a \(6\) foot fence and at \(15\) foot board. The sun is shining directly overhead. You want to lean your board against the fence so that as large a shadow as possible covers your neighbor's yard. What is the maximum length of the shadow you can form? The ladder must rest against the fence and touch the ground on your side of the fence.

Solution

让一个木板斜靠在一个栅栏上，木板的一侧得靠在自己一段。问在邻居那侧的阴影长度（最大化）。

利用相似，自己那侧的阴影长度为 \(x\), 邻居那侧为 \(y\)：

\[\frac{x}{\sqrt{x^2+36}}=\frac{x+y}{15} \]

反解出 \(y\) 以后，令 \(y'=0\).