Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

검증된 단계별 안내

Understand that the problem asks for the smaller angle between the hour and minute hands of a clock at 3:15.

Calculate the position of the minute hand: since the minute hand moves 360 degrees in 60 minutes, its angle from the 12 o'clock position is given by \(\text{Minute Angle} = 6 \times \text{minutes}\). For 15 minutes, this is \(6 \times 15\) degrees.

Calculate the position of the hour hand: the hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. Additionally, it moves 0.5 degrees per minute. The formula for the hour hand angle is \(\text{Hour Angle} = 30 \times \text{hours} + 0.5 \times \text{minutes}\). For 3:15, substitute hours = 3 and minutes = 15.

Find the difference between the two angles calculated: \(\text{Angle Difference} = |\text{Hour Angle} - \text{Minute Angle}|\).

Since the clock is circular, the smaller angle between the hands is the minimum of \(\text{Angle Difference}\) and \(360 - \text{Angle Difference}\). This gives the measure of the smaller angle formed at 3:15.

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주요 개념

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Angle Measurement of Clock Hands

The position of clock hands can be translated into angles measured from the 12 o'clock position. The minute hand moves 6 degrees per minute (360°/60), while the hour hand moves 0.5 degrees per minute (30° per hour). Understanding these rates is essential to calculate the exact angles at any given time.

Calculating the Angle Between Two Lines

The angle between two clock hands is found by taking the absolute difference of their individual angles from the 12 o'clock position. Since the clock is circular, if this difference exceeds 180 degrees, the smaller angle is found by subtracting it from 360 degrees.

Time to Angle Conversion for the Hour Hand

Unlike the minute hand, the hour hand moves continuously as time passes. At 3:15, the hour hand is not exactly at 3 but has moved a quarter of the way towards 4. This continuous movement is calculated by adding 0.5 degrees per minute to the base hour angle.