Question

In a day how many times the minute-hand and hour-hand make right angle between them? (a) 12 (b) 20 (c) 23 (d) 44

Short answer

Answer: In a day (24 hours), the minute-hand and hour-hand of a clock form a right angle 44 times.

Step-by-step solution

Understand the relative speeds of the minute-hand and hour-hand

The minute-hand completes one full cycle (360 degrees) in 60 minutes, which means its speed is 360 degrees per hour. The hour-hand completes a full cycle (360 degrees) in 12 hours, which means its speed is 30 degrees per hour.

Calculate the relative speed between the minute-hand and hour-hand

Subtract the speed of the hour-hand from the speed of the minute-hand to get the relative speed. In this case, the relative speed is 360 - 30 = 330 degrees per hour.

Determine when a right angle is formed between the hour-hand and minute-hand

A right angle occurs when the angle between the hands is 90 degrees (or 270 degrees, since it's a symmetric case). To find when they form a 90-degree angle, we need to find the time it takes for the relative speed to equal 90 degrees, which is the time for their relative position to change 90 degrees. Therefore, we can use the formula: Time = (Change in angle)/(Relative speed) We have two cases: Case 1: When the change in angle is 90 degrees, the time is t1 = 90/330 hours. Case 2: When the change in angle is 270 degrees (the symmetric case), the time is t2 = 270/330 hours.

Count the occurrences of right angles in 12 hours

To find the number of times they form a right angle in 12 hours, we should note that within one hour, a right angle is formed twice (once in case 1 and once in case 2). However, when the minute-hand and hour-hand are at 12 (minute = 0 and hour = 12), they don't form a right angle. So, in 12 hours, there are 11 hours where the right angle is formed twice, which means 11 * 2 = 22 total occurrences of right angles.

Multiply the occurrences by 2 to find the number of right angles in 24 hours

Since there are 22 occurrences of right angles in 12 hours, there will be twice as many occurrences in 24 hours: Number of right angles in 24 hours = 22 * 2 = 44

Choose the correct answer

Based on our calculations, the minute-hand and hour-hand form a right angle 44 times in a day (24 hours). So, the correct answer is (d) 44.

Key concepts

Relative Speed of Clock Hands

The concept of relative speed between the clock hands is crucial for solving problems related to angles between them. The minute-hand and hour-hand move at different speeds around the clock face. The minute-hand completes a full circle of 360 degrees every hour, which translates to a speed of 360 degrees per hour. Meanwhile, the hour-hand takes 12 hours to complete the same 360-degree circle, so its speed is much slower, at 30 degrees per hour.

To determine how often a specific angle is formed, such as a right angle, we need to calculate the relative speed between these two hands. This is done by subtracting the speed of the hour-hand from the speed of the minute-hand: 360 - 30 = 330 degrees per hour. This means that, in one hour, the minute-hand travels 330 degrees more than the hour-hand. Understanding this difference in speed is key to calculating the time intervals when the hands align at particular angles.

Angle Measurement in Clocks

In clock angle problems, identifying the exact angle between the clock hands at a given time is the main task. Specifically, a right angle is one of the most common angles to consider. A right angle measures 90 degrees. On a clock, the hands form a right angle twice every hour: once at a 90-degree position and the other at a 270-degree position, which is essentially the same angle measured counterclockwise.

The clock face is divided into 12 hours, each corresponding with a 30-degree segment (since 360 degrees / 12 hours = 30 degrees per hour). To determine the position of the hands at any time:

• The minute-hand at any minute is 6 degrees for each minute past 12. For instance, at 15 minutes past, it is at 90 degrees (15*6).
• The hour-hand moves gradually and is 0.5 degrees per minute (30 degrees per hour / 60 minutes).

By calculating how much each hand has moved, the angle between them can be calculated by evaluating the absolute difference between the angles of the minute and hour hands. Keeping an understanding of these angular movements facilitates the solution of clock-related problems.

Time Calculation Using Angles

To figure out when the clock hands form a certain angle, like a right angle, it's necessary to understand and use formulas. Specifically, the formula for determining time based on angle involves the relative movement of the clock hands. The formula is:
\[\text{Time} = \frac{\text{Change in Angle}}{\text{Relative Speed}}\]
For a right angle (90 degrees) and a full hour (330 degrees per hour), two scenarios are considered:

• First, the 90-degree position is reached. This takes: \( t_1 = \frac{90}{330} \) hours.
• The second right angle is the complementary 270-degree position (since 360 - 90 = 270), which takes: \( t_2 = \frac{270}{330} \) hours.

Both these calculations reveal how often these angles form in a cycle. Since two right angles occur per hour, but one hour at 12 o'clock doesn’t count, there are 22 occurrences in 12 hours. To find the total for 24 hours, simply multiply by two, yielding 44 right angles each day. This understanding aids in managing clock-related time calculations and solving similar problems efficiently.