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Math

QuestionFind the smaller angle between clock hands at 5:25.

Studdy Solution

STEP 1

Assumptions1. We are looking at a standard12-hour analog clock. . The hour hand moves continuously, not just at the start of a new hour.
3. We are looking for the smaller angle formed by the hour and minute hands at525.

STEP 2

First, we need to calculate the position of the hour hand. At the start of the hour (500), the hour hand is at the5 on the clock, which is150 degrees from the12. The hour hand moves0.5 degrees per minute.
Hour hand position=Start position+(Minutes past the hourtimesDegrees per minute)Hour\, hand\, position = Start\, position + (Minutes\, past\, the\, hour \\times Degrees\, per\, minute)

STEP 3

Now, plug in the given values for the start position (150 degrees), minutes past the hour (25), and degrees per minute (0.5) to calculate the hour hand position.
Hour hand position=150 degrees+(25 minutestimes0.5 degrees/minute)Hour\, hand\, position =150\, degrees + (25\, minutes \\times0.5\, degrees/minute)

STEP 4

Calculate the hour hand position.
Hour hand position=150 degrees+(25 minutestimes0. degrees/minute)=162. degreesHour\, hand\, position =150\, degrees + (25\, minutes \\times0.\, degrees/minute) =162.\, degrees

STEP 5

Next, we need to calculate the position of the minute hand. The minute hand moves degrees per minute.
Minute hand position=Minutes past the hourtimesDegrees per minuteMinute\, hand\, position = Minutes\, past\, the\, hour \\times Degrees\, per\, minute

STEP 6

Now, plug in the given values for the minutes past the hour (25) and degrees per minute (6) to calculate the minute hand position.
Minute hand position=25 minutestimes6 degrees/minuteMinute\, hand\, position =25\, minutes \\times6\, degrees/minute

STEP 7

Calculate the minute hand position.
Minute hand position=25 minutestimes6 degrees/minute=150 degreesMinute\, hand\, position =25\, minutes \\times6\, degrees/minute =150\, degrees

STEP 8

Now that we have the positions of the hour and minute hands, we can find the angle between them by subtracting the smaller angle from the larger one.
Angle=∣Hour hand position−Minute hand position∣Angle = |Hour\, hand\, position - Minute\, hand\, position|

STEP 9

Plug in the values for the hour hand position and the minute hand position to calculate the angle.
Angle=∣162.5 degrees−150 degrees∣Angle = |162.5\, degrees -150\, degrees|

STEP 10

Calculate the angle.
Angle=∣162.5 degrees−150 degrees∣=12.5 degreesAngle = |162.5\, degrees -150\, degrees| =12.5\, degreesThe smaller angle formed by the hands of a clock at525 is12.5 degrees.

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