3B. On a square plot of land with a side of $12 \mathrm{~m}$, a cylindrical pit with a diameter (diameter of the base) of $8 \mathrm{~m}$ is dug. The excavated earth is evenly distributed over the remaining part of the plot and is compacted as much as it was when it was in the pit. How deep should it be dug to make the pit $3 \mathrm{~m}$ deep?
Solution. The area of the land where the excavated soil from the pit is distributed is

$$
\left(12^{2}-4^{2} \pi\right) \mathrm{m}^{2}=(144-16 \pi) \mathrm{m}^{2}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_ce8ee69309f5a5d5bef7g-02.jpg?height=346&width=908&top_left_y=1804&top_left_x=416)

Let $x$ be the depth of the excavation, then the thickness of the layer of compacted soil should be $(3-x) \mathrm{m}$. Since the excavated soil is compacted as much as it was when it was in the pit, this means that the volume of the excavated soil is equal to the volume of the uniformly spread and compacted soil on the remaining part of the land, i.e.,

$$
4^{2} \pi \cdot x=(144-16 \pi) \cdot(3-x)
$$

By solving the above equation for $x$, we get $x=\left(3-\frac{\pi}{3}\right) \mathrm{m}$. Since $\pi \approx 3.14$, it follows that the excavation depth should be $x \approx 1.95 \mathrm{~m}$.