Question

The hour hand of an analog clock moves in proportion to the movement of the minute hand. This means that at \(4: 05,\) the hour hand will have moved beyond the 4 by \(\frac{5}{60}\) of the distance it would move in an hour. a) What is the measure of the obtuse angle between the hands of a clock at \(4: 00 ?\) Give your answer in degrees. b) What is the measure, in degrees, of the acute angle between the hands of a clock at \(4: 10 ?\) c) At certain times, the hands of a clock are at right angles to each other. What are two of these times? d) At how many different times does the angle between the hands of a clock measure \(90^{\circ}\) between 4: 00 and \(5: 00 ?\) e) Does one of the times occur before, at, or shortly after \(4: 05 ?\) Explain.

Short answer

a) 240 degrees; b) 65 degrees; c) About 4:05.5 and 4:38.2; d) 2 times; e) Shortly after 4:05.

Step-by-step solution

Calculate the Angle at 4:00

The minute hand is at the 12, and the hour hand is at the 4. Each hour mark on the clock represents 30 degrees because \[ \frac{360^{\circ}}{12} = 30^{\circ} \] Therefore, the angle at 4:00 is \[ 4 \times 30^{\circ} = 120^{\circ} \] The obtuse angle is the larger angle on the clock, which would be \[ 360^{\circ} - 120^{\circ} = 240^{\circ} \]

Find the Angle at 4:10

At 4:10, the minute hand has moved to the 2 (10 minutes past 12), which is \[ 10 \times 6^{\circ} = 60^{\circ} \] The hour hand has moved slightly past 4. Since each hour is 30 degrees and there are 60 minutes in an hour, the movement of the hour hand can be computed as: \[ 4 \times 30^{\circ} + \left( \frac{10}{60} \right) \times 30^{\circ} = 120^{\circ} + 5^{\circ} = 125^{\circ} \] The difference between the two positions is: \[ 125^{\circ} - 60^{\circ} = 65^{\circ} \] Since this is an acute angle, the measure is \[ 65^{\circ} \]

Find the Times When Hands are Perpendicular

The formula to find the times when the clock hands are 90 degrees apart is: \[ \left| 30H - 5.5M \right| = 90^{\circ} \] where H is the hour, and M is the minute. For the hands to be perpendicular: \[ 30 \times 4 - 5.5M = 90 \] or \[ 5.5M - 30 \times 4 = 90 \] Solving the first equation, \[ 120 - 5.5M = 90 \Longrightarrow 5.5M = 30 \Longrightarrow M = \frac{30}{5.5} \approx 5.45 \] Solving the second equation, \[ 5.5M = 210 \Longrightarrow M = \frac{210}{5.5} \approx 38.18 \] Hence, two times are possible: about 4:05.5 and 4:38.2

Find Number of Times 90 Degrees Between 4:00 and 5:00

From Step 3, only two times exist when the minute and hour hands create a 90-degree angle between 4:00 and 5:00. Thus, there are 2 such times.

Determine if One Time is Shortly After 4:05

One of the times when the clock hands are at a 90-degree angle is just after 4:05 (approximately 4:05.5). Thus, this occurs shortly after 4:05.

Key concepts

angle calculation

Understanding how to calculate angles on a clock is crucial for solving time-related problems. The most common angles include acute, obtuse, and right angles.
To start, each hour mark on an analog clock represents 30 degrees.
To find the angle between the hands, you can use this formula:

• Angle = \(|30H - 5.5M|\)

where H is the hour and M is the minute.
For example, at 4:00, the hour hand is at 120 degrees from the 12 o'clock position (4 hours * 30 degrees).

clock problems

Clock problems often test your understanding of how the hour and minute hands move. They may ask you to find specific angles or determine when the hands form certain angles.
When solving these problems, remember

• The minute hand moves 6 degrees per minute (360 degrees/60 minutes).
• The hour hand moves 0.5 degrees per minute (30 degrees/60 minutes).

These points are key to figuring out the exact positions of the hands at given times. Understanding these movements can make solving clock problems much easier.

right angles

Right angles on a clock occur when the hands form a 90-degree angle. These angles are critical in many clock-related problems.
You can find times when the hands form right angles using the formula:

• \(|30H - 5.5M| = 90\)

By solving this equation, you can pinpoint the exact minutes when the hands are perpendicular. For instance, between 4:00 and 5:00, right angles occur at about 4:05.5 and 4:38.2.

analog clocks

Analog clocks have circular faces and hands that rotate to show time. The basic structure includes 12 hour marks and 60 minute marks.
Understanding how to read these clocks and calculate angles between hands is essential. Remember:

• The hour hand moves gradually, covering 30 degrees per hour.
• The minute hand moves quickly, covering 360 degrees per hour.

Accurate angle calculation involves considering these movements for both hands.

obtuse angle

An obtuse angle is greater than 90 degrees but less than 180 degrees. These angles on a clock are quite noticeable.
To find an obtuse angle, you generally need to calculate the total degrees first and then subtract the smaller angle from 360 degrees. For example, the angle at 4:00 is 120 degrees, making the obtuse angle 240 degrees (360 - 120).
This is crucial for understanding larger angles between clock hands.

acute angle

An acute angle is less than 90 degrees. These angles occur frequently on clocks and require precise calculation.
To determine an acute angle, you find the smaller angle between the hour and minute hands. For example, at 4:10, the acute angle is 65 degrees. You calculate this using:

• Minute hand position: 10 * 6 degrees = 60 degrees
• Hour hand position: 4 * 30 degrees + (10/60) * 30 degrees = 125 degrees
• Difference: |125 - 60| = 65 degrees

This method ensures you accurately identify the acute angle.