901. Well Drilling

A driller drills for water. At each iteration the driller chooses a depth $d$$d$ (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby.

Drilling to depth $d$$d$ takes exactly $d$$d$ hours. The groundwater depth is constant in the relevant area and its distribution is known to be an exponential random variable with expected value of $1$$1$. In other words, the probability that the groundwater is deeper than $d$$d$ is $e^{-d}$$e^{-d}$.

Assuming an optimal strategy, find the minimal expected drilling time in hours required to find water. Give your answer rounded to 9 places after the decimal point.

901. 钻井

一位钻井工人正在钻井。在每次，钻井工人任选一个正实数 $d$$d$，钻井至距地表深度为 $d$$d$ 的地方，并检查是否能找到水。如果没能找到水，钻井工人将回到地表，选择一处新地方并重复执行上述过程；否则，钻井过程即刻终止。

钻井至距地表深度为 $d$$d$ 的地方需要 $d$$d$ 小时。在钻井区域内，地下水深度恒定，其分布服从期望是 $1$$1$ 的 指数分布。换句话说，地下水的深度 $>d$$> d$ 的概率是 ${\mathrm{e}}^{-d}$$\mathrm{e}^{-d}$。

若钻井工人采取最优策略，请问钻井工人为找到地下水，期望需要钻井多少小时？将你的答案四舍五入至小数点后第 9 位。

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