$\"\"$

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(I)

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Radius of the pulley is $\"\"$ inches.

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Angle of each rotation is $\"\"$ .

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Angle of $\"\"$ revolution is $\"\"$ radians.

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Linear speed $\"\"$ .

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

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$\"\"$

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$\"\"$ in/min.

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$\"\"$

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(II)

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Radius of the pulley is $\"\"$ inches.

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Angle of each rotation is $\"\"$ .

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Angle of $\"\"$ revolution is $\"\"$ radians.

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

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$\"\"$

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$\"\"$ in/min.

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$\"\"$

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(III)

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Radius of the pulley is $\"\"$ inches.

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Angle of each rotation is $\"\"$ .

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Angle of $\"\"$ revolution is $\"\"$ radians.

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

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$\"\"$

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$\"\"$ in/min.

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Pulley has the highest linear speed in Scenario III.

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Scenario III would be the fastest method to use to lift a distance of 15 feet fastest.

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$\"\"$

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Scenario III would be the fastest method to use to lift a distance of 15 feet fastest.