• Choice G
    56. As shown below, 2 circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger pulley, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches?

    ME = AM – AE = AM – BN = 5 – 1 = 4;

    MN = 8

    ⇒ AB = CD = EN = √(ME² + MN²) = 4√3

    arc AMD = 2/3 ⋅ 2π ⋅ 5 = 20/3 ⋅ π

    arc BNC = 1/3 ⋅ 2π ⋅ 1 = 2/3 ⋅ π

    ⇒ Length of the belt = arc AMD + AB + arc BNC + CD = 20/3 ⋅ π + 4√3 + 2/3 ⋅ π + 4√3 = 22/3 ⋅ π + 8√3

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