Final answer:
To find the diameter of Big Ben's clock given the arc length of 3.67 meters from 12 to 2, the arc length formula must be rearranged to solve for the diameter, considering the angle that corresponds to the 2 hours on the clock face.
Step-by-step explanation:
To calculate the diameter of the clock on Big Ben given the arc length from 12 to 2, which measures 3.67 meters, we must first understand the relationship between the arc length, the angle subtended by the arc, and the clock's radius. Since this is a standard clock and the arc represents 2 hours out of the 12 hours of the clock's full cycle, the angle subtended is 1/6 of a full circle, or approximately 60 degrees (or π/3 radians).
The formula relating the arc length (s), radius (r), and the angle in radians (θ) is s = rθ. We can rearrange this formula to solve for the diameter by first finding the radius: r = s/θ. As the diameter is twice the radius, the formula for the diameter (d) is d = 2s/θ. With a 60-degree angle corresponding to π/3 radians, we calculate the diameter by multiplying the given arc length by 2 and dividing by π/3:
d = 2(3.67 m) / (π/3) = 7.34 m / (π/3). This calculation will yield the diameter of the clock on Big Ben.