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Chapter 4: Problem 17

What is the angle between the hour hand and the minute hand on a clock at \(4: 30 ?\)

Short Answer

Expert verified
The angle between the hour hand and the minute hand at 4:30 is \(45\) degrees.

Step by step solution

01

Calculate the position of the hour hand

First, let's find out the position of the hour hand. In 12 hours, the hour hand moves 360 degrees, which means it moves 360/12 = 30 degrees per hour. At 4:30, the hour hand is 4.5 hours (4 hours and 30 minutes) away from the 12-hour position. Multiply 4.5 hours by 30 degrees per hour to find the position of the hour hand: \( 4.5 \cdot 30 = 135 \) degrees.
02

Calculate the position of the minute hand

Now, let's find out the position of the minute hand. In an hour (60 minutes), the minute hand moves 360 degrees, which means it moves 360/60 = 6 degrees per minute. At 4:30, the minute hand is 30 minutes away from the 12-hour position. Multiply 30 minutes by 6 degrees per minute to find the position of the minute hand: \( 30 \cdot 6 = 180 \) degrees.
03

Find the angle between the hour and minute hands

Now that we have the positions of the hour hand and the minute hand, we can find the angle between them. If we subtract the smaller angle from the larger angle, we will find the angle between the two hands: \( |180 - 135| = 45 \) degrees. So, the angle between the hour hand and the minute hand at 4:30 is 45 degrees.

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