Question: A monkey is swinging from a tree. On the first swing, she passes through an arc of $20$ m. With each swing, she passes through an arc of $\frac{4}{5}$ the length of the previous swing.
What is the total distance the monkey has traveled when she completes her 10th swing? Round your final answer to the nearest meter.
1. **State the problem:**
A monkey swings from a tree. The first swing covers an arc length of $20$ m. Each subsequent swing covers $\frac{4}{5}$ of the previous swing's arc length. We want to find the total distance traveled after $10$ swings.
2. **Identify the type of sequence:**
The arc lengths form a geometric sequence where:
- First term $a = 20$
- Common ratio $r = \frac{4}{5}$
3. **Formula for the sum of the first $n$ terms of a geometric sequence:**
$$ S_n = a \frac{1 - r^n}{1 - r} $$
where $S_n$ is the sum of the first $n$ terms.
4. **Apply the formula for $n=10$:**
$$ S_{10} = 20 \times \frac{1 - \left(\frac{4}{5}\right)^{10}}{1 - \frac{4}{5}} $$
5. **Simplify the denominator:**
$$ 1 - \frac{4}{5} = \frac{1}{5} $$
6. **Rewrite the sum:**
$$ S_{10} = 20 \times \frac{1 - \left(\frac{4}{5}\right)^{10}}{\frac{1}{5}} $$
7. **Dividing by $\frac{1}{5}$ is multiplying by $5$: **
$$ S_{10} = 20 \times 5 \times \left(1 - \left(\frac{4}{5}\right)^{10}\right) $$
8. **Calculate $\left(\frac{4}{5}\right)^{10}$:**
$$ \left(\frac{4}{5}\right)^{10} = \frac{4^{10}}{5^{10}} = \frac{1048576}{9765625} \approx 0.1074 $$
9. **Calculate the sum:**
$$ S_{10} = 100 \times (1 - 0.1074) = 100 \times 0.8926 = 89.26 $$
10. **Round to the nearest meter:**
$$ \boxed{89} \text{ meters} $$
**Final answer:** The monkey has traveled approximately $89$ meters after completing the 10th swing.