$\"\"$ \ \

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Initlally the length of the arc is $\"\"$ feet. \ \

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On each successive swing, the length of the arc is $\"\"$ of the previous length. \ \

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The first term of the geometric sequence is $\"\"$ . \ \

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The common ratio is $\"\"$ . \ \

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(a) \ \

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Formula for $\"\"$ term in geometric sequence is $\"\"$ . \ \

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Substitute $\"\"$ and $\"\"$ . \ \

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$\"\"$ . \ \

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Find the length of the arc of the $\"\"$ swing. \ \

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Substitute $\"\"$ in $\"\"$ . \ \

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$\"\"$ \ \

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$\"\"$ . \ \

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Length of the arc of the $\"\"$ swing is $\"\"$ feet. \ \

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$\"\"$ \ \

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(b) \ \

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Find for which swing is the length of the arc less than $\"\"$ foot. \ \

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$\"\"$ \ \

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$\"\"$ \ \

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Take logarithm of each side. \ \

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$\"\"$ \ \

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Therefore, the length of the swing is less than $\"\"$ foot on the $\"\"$ swing. \ \

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$\"\"$ \ \

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(c) \ \

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Formula for total length of the pendulum swung after $\"\"$ swings. \ \

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Sum of the first $\"\"$ terms of the geometric series, $\"\"$ . \ \

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Where $\"\"$ is the first term and $\"\"$ is the common ratio. \ \

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Substitute $\"\"$ and $\"\"$ in $\"\"$ . \ \

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$\"\"$ \ \

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Substitute $\"\"$ . \ \

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$\"\"$ \ \

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$\"\"$ . \ \

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Total length of the pendulum swung after $\"\"$ swings is $\"\"$ feet. \ \

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$\"\"$ \ \

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(d) \ \

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Find the total length of the pendulum after coimg to stop. \ \

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If $\"\"$ , the infinite series conveges and its sum is $\"\"$ . \ \

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Where $\"\"$ is the first term and $\"\"$ is the common ratio. \ \

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Substitute $\"\"$ and $\"\"$ in $\"\"$ . \ \

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$\"\"$ \ \

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Total length of the pendulum swung after coming to stop is $\"\"$ feet. \ \

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$\"\"$ \ \

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(a) Length of the arc of the $\"\"$ swing is $\"\"$ feet. \ \

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(b) The length of the swing is less than $\"\"$ foot on the $\"\"$ swing. \ \

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(c) Total length of the pendulum swung after $\"\"$ swings is $\"\"$ feet. \ \

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(d) Total length of the pendulum swung after coming to stop is $\"\"$ feet.