A monkey is swinging from a tree. On each swing, she travels along an arc that is 75% as long as the previous swing's arc. The total length of the arcs from her first 4 swings is 175m. How long was the monkey's 1^("st ") swing? Round your final answer to the nearest meter. m

Question

A monkey is swinging from a tree. On each swing, she travels along an arc that is  75% as long as the previous swing's arc. The total length of the arcs from her first 4 swings is  175m. How long was the monkey's  1^("st ") swing? Round your final answer to the nearest meter. m

Bytelearn · Accepted Answer

Denote Length: Let's denote the length of the first swing as x meters. According to the problem, each subsequent swing is 75% the length of the previous one. So, the second swing will be 0.75x, the third swing will be 0.75^2 * x, and the fourth swing will be 0.75^3 * x. We are given that the total length of the arcs from her first 4 swings is 175 meters. We can set up the following equation to represent this relationship: x + 0.75x + (0.75^2 * x) + (0.75^3 * x) = 175Set Up Equation: Now we need to solve for x. First, let's simplify the equation by combining like terms: x * (1 + 0.75 + 0.75^2 + 0.75^3) = 175Simplify Equation: Calculate the sum of the geometric series in the parentheses: 1 + 0.75 + 0.75^2 + 0.75^3 = 1 + 0.75 + 0.5625 + 0.421875Calculate Series Sum: Add up the numbers from the previous step to get the sum of the series: 1 + 0.75 + 0.5625 + 0.421875 = 2.734375Divide and Solve: Now, divide both sides of the equation by the sum of the series to solve for x: x = 175 / 2.734375Perform the division to find the length of the first swing: x ≈ 175 / 2.734375 ≈ 63.99913 Since we need to round to the nearest meter, x ≈ 64 meters.

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