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A monkey is swinging from a tree. On the first swing, she passes through an arc of $20\text{m}$ . With each swing, she passes through an arc $\frac{4}{5}$ the length of the previous swing. What is the total distance the monkey has traveled when she completes her $10^{\text{th}}$ swing? Round your final answer to the nearest meter.

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Circles: word problems

Full solution

Q. A monkey is swinging from a tree. On the first swing, she passes through an arc of

20\text{m}

. With each swing, she passes through an arc

\frac{4}{5}

the length of the previous swing. What is the total distance the monkey has traveled when she completes her

10^{\text{th}}

swing? Round your final answer to the nearest meter.

Calculate distance using formula: Calculate the distance of each swing using the geometric sequence formula. $\newline$ First swing = $20m$ , $\newline$ Second swing = $(\frac{4}{5}) \times 20m = 16m$ , $\newline$ Third swing = $(\frac{4}{5}) \times 16m = 12.8m$ , and so on.
Use geometric series formula: Use the formula for the sum of a geometric series: $S_n = a \times (1 - r^n) / (1 - r)$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ Here, $a = 20\,\text{m}$ , $r = \frac{4}{5}$ , and $n = 10$ . $\newline$ $S_{10} = 20 \times (1 - (\frac{4}{5})^{10}) / (1 - \frac{4}{5})$
Calculate $(4/5)^{10}$ : Calculate $(4/5)^{10}$ using a calculator. $\newline$ $(4/5)^{10} \approx 0.1074$
Substitute into formula: Substitute back into the sum formula. $\newline$ $S_{10} = 20 \times (1 - 0.1074) / (1 - \frac{4}{5})$ $\newline$ $S_{10} = 20 \times 0.8926 / 0.2$ $\newline$ $S_{10} = 89.26 / 0.2$ $\newline$ $S_{10} = 446.3$
Round final answer: Round the final answer to the nearest meter. $\newline$ Final distance $\approx 446$ meters.

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