Question

Find the measure of the smaller angle formed by the hands of a clock at the following times.

Accepted Answer

Understand that the problem asks for the smaller angle between the hour and minute hands of a clock at a given time. The clock is circular, so the angle between the hands can be found by calculating the positions of each hand and then finding the difference. Calculate the position of the minute hand in degrees. Since the minute hand moves 360 degrees in 60 minutes, its position at $$ m $$ minutes is given by the formula:

$$	ext{Minute angle} = 6 	imes m$$ Calculate the position of the hour hand in degrees. The hour hand moves 360 degrees in 12 hours, or 30 degrees per hour, plus it moves 0.5 degrees per minute (because it moves continuously). For $$ h $$ hours and $$ m $$ minutes, the formula is:

$$	ext{Hour angle} = 30 	imes h + 0.5 	imes m$$ Find the absolute difference between the hour and minute angles:

$$	ext{Angle difference} = |	ext{Hour angle} - 	ext{Minute angle}|$$ Since the clock is circular, the smaller angle between the hands is the minimum of the angle difference and its supplement to 360 degrees:

$$	ext{Smaller angle} = \min(	ext{Angle difference}, 360 - 	ext{Angle difference})$$

Find the measure of the smaller angle formed by the hands of a clock at the following times.

Verified video answer for a similar problem:

Key Concepts

Angle Measurement in Clocks

Calculating Hour and Minute Hand Positions

Finding the Smaller Angle Between Two Hands