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

A monkey is swinging from a tree. On the first swing, she passes through an arc of 20 m 20 \mathrm{~m} . With each swing, she passes through an arc 45 \frac{4}{5} the length of the previous swing.\newlineWhat is the total distance the monkey has traveled when she completes her 10th  10^{\text {th }} swing?\newlineRound your final answer to the nearest meter.\newline□ \square m \mathrm{m}

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Q. A monkey is swinging from a tree. On the first swing, she passes through an arc of 20 m 20 \mathrm{~m} . With each swing, she passes through an arc 45 \frac{4}{5} the length of the previous swing.\newlineWhat is the total distance the monkey has traveled when she completes her 10th  10^{\text {th }} swing?\newlineRound your final answer to the nearest meter.\newline□ \square m \mathrm{m}
  1. Identify Swing Lengths: Identify the sequence of the lengths of the swings.\newlineThe monkey swings through an arc of 20m20m on the first swing. Each subsequent swing is (4/5)(4/5) times the length of the previous swing. This creates a geometric sequence for the lengths of the swings.\newlineFirst swing length = 20m20m\newlineSecond swing length = 20m×(4/5)20m \times (4/5)\newlineThird swing length = 20m×(4/5)220m \times (4/5)^2\newline...\newlineTenth swing length = 20m×(4/5)920m \times (4/5)^9
  2. Calculate Total Distance: Calculate the total distance traveled by summing the lengths of the swings.\newlineThe total distance is the sum of the first 1010 terms of the geometric sequence.\newlineTotal distance = 20m+20m∗(45)+20m∗(45)2+…+20m∗(45)920m + 20m*(\frac{4}{5}) + 20m*(\frac{4}{5})^2 + \ldots + 20m*(\frac{4}{5})^9
  3. Use Geometric Sequence Formula: Use the formula for the sum of the first nn terms of a geometric sequence.\newlineThe sum of the first nn terms of a geometric sequence is given by:\newlineSn=a×(1−rn)/(1−r)S_n = a \times (1 - r^n) / (1 - r), where aa is the first term, rr is the common ratio, and nn is the number of terms.\newlineHere, a=20ma = 20m, r=(4/5)r = (4/5), and n=10n = 10.\newlineS10=20m×(1−(4/5)10)/(1−(4/5))S_{10} = 20m \times (1 - (4/5)^{10}) / (1 - (4/5))
  4. Calculate Sum: Calculate the sum using the values for a, r, and n.\newlineS10=20m×(1−(45)10)/(1−(45))S_{10} = 20m \times (1 - (\frac{4}{5})^{10}) / (1 - (\frac{4}{5}))\newlineS10=20m×(1−(0.1048576))/(0.2)S_{10} = 20m \times (1 - (0.1048576)) / (0.2)\newlineS10=20m×(0.8951424)/(0.2)S_{10} = 20m \times (0.8951424) / (0.2)\newlineS10=20m×4.475712S_{10} = 20m \times 4.475712\newlineS10=89.51424mS_{10} = 89.51424m
  5. Round Final Answer: Round the final answer to the nearest meter.\newlineThe total distance traveled, rounded to the nearest meter, is approximately 9090 meters.

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