The forest's timber stock is estimated at $30500 \mathrm{~m}^{3}$; its annual growth is estimated at $2 \%$. What will the timber stock be after 11 years if $1400 \mathrm{~m}^{3}$ of wood is cut down at the end of each year? How many years would it take for the forest's timber stock to be completely depleted in this manner?
If we denote the original forest stock by $v_{0}$, and the annual growth is $p \%$, then the forest stock at the end of the $n$-th year is

$$
v_{n}=v_{0} q^{n}, \quad \text { where } \quad q=1+\frac{p}{100}
$$

However, since a constant amount $r$ of wood is cut down at the end of each year, this amount (including the growth on the cut wood) must be subtracted. This sum is

$$
r \cdot q^{n-1}+r \cdot q^{n-2}+\ldots r q+r=r \frac{q^{n}-1}{q-1}
$$

Therefore, the forest stock at the end of the $n$-th year is

$$
V_{n}=v_{0} q^{n}-r \frac{q^{n-1}}{q-1}
$$

In this case,

$$
\begin{gathered}
V_{11}=30500 \cdot 1,02^{11}-1400 \frac{1,02^{11}-1}{0,02}=30500 \cdot 1,02^{11}-70000 \cdot\left(1,02^{11}-1\right)= \\
=70000-39500 \cdot 1,02^{11}=70000-A
\end{gathered}
$$

Using the school's four-digit logarithm table:

![](https://cdn.mathpix.com/cropped/2024_05_02_faacaa2eb6e952fa8f45g-1.jpg?height=180&width=430&top_left_y=978&top_left_x=837)

Thus, the forest stock at the end of the 11th year is

$$
v_{11}=70000-49110=20890 \mathrm{~m}^{3}
$$

The depletion of the forest stock will occur $x$ years later, when

$$
70000-39500 \cdot 1,02^{x}=0
$$

or

$$
395 \cdot 1,02^{x}=700
$$

from which

$$
x=\frac{\lg 700-\lg 395}{\lg 1,02}=\frac{2,8451-2,5966}{0,0086}=\frac{2485}{86} \approx 28,9
$$

which means that the forest stock will be depleted in the 29th year, but there will not be enough for the final year's 1400 m³ cut.