The minute hand on the clock atop city hall measures 6 feet...

The minute hand on the clock atop city hall measures 6 feet 3 inches from its tip to its axle. a. Through what angle (in radians) does the minute hand pass between 9: 12 A.M. and 9: 48 A.M.? $\frac{6 \pi}{5}$ radians b. What distance, to the nearest tenth of a foot, does the tip of the minute hand travel during this period? $23.6 \mathrm{ft}$

The minute hand on the clock atop city hall measures 6 feet 3 inches from its tip to its axle.
a. Through what angle (in radians) does the minute hand pass between 9: 12 A.M. and 9: 48 A.M.? $\frac{6 \pi}{5}$ radians
b. What distance, to the nearest tenth of a foot, does the tip of the minute hand travel during this period? $23.6 \mathrm{ft}$

Key Concepts

Arc Length Calculation

The arc length of a circle is determined by the product of the radius and the angular displacement in radians. This relationship, expressed in the formula s = r?, is fundamental in linking the linear distance traveled by a point on the rim of a rotating object with its angular movement. It is widely applicable in solving practical problems involving circular paths.

Unit Conversion

Unit conversion is an essential step in solving problems where measurements may be given in different units. It usually entails converting quantities into consistent units before performing calculations. In this context, converting measurements such as lengths (for instance, inches to feet) ensures accuracy in applying mathematical formulas such as the one for arc length.

Angular Displacement in Radians

Angular displacement is the measure of the angle through which a point or line has been rotated in a specified sense about a specified axis. In many problems involving circular motion, especially those relating to clocks, the angle of rotation is naturally expressed in radians, with one full revolution equaling 2? radians. This concept is central when calculating how much a rotating object moves within a given time frame.

Proportional Reasoning for Time and Angle

This concept involves relating a fraction of a complete cycle to the corresponding angular displacement. For example, if a full rotation takes a certain amount of time (like 60 minutes for a clock), then any given interval of time represents a proportional part of that full angle. This method is used to determine the specific angle traversed in scenarios involving uniform rotational motion.

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