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

Question

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

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Identify initial arc length: Identify the initial length of the arc for the first swing and the pattern of the decrease in the length of the arc for each subsequent swing. The initial length of the arc for the first swing is 20 meters. The length of each subsequent swing is (4/5) times the length of the previous swing.Calculate total distance: Calculate the total distance traveled by the monkey over the 10 swings using the formula for the sum of a geometric series. The sum of a geometric series is given by S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms. Here, a_1 = 20m, r = 4/5, and n = 10.Plug values into formula: Plug the values into the formula to calculate the sum of the first 10 terms of the geometric series. S_10 = 20 * (1 - (4/5)^10) / (1 - 4/5)Simplify expression to find sum: Simplify the expression to find the sum S_10. S_10 = 20 * (1 - (4/5)^10) / (1/5) Calculate (4/5)^10 using a calculator. (4/5)^10 ≈ 0.1073741824 Now, substitute this value into the expression. S_10 = 20 * (1 - 0.1073741824) / (1/5) S_10 = 20 * (0.8926258176) / (1/5) S_10 = 20 * (0.8926258176) * 5 S_10 = 20 * 4.463129088 S_10 = 89.26258176Round final answer: Round the final answer to the nearest meter. S_10 ≈ 89 meters

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