At what time between 4:00 and 5:00 will the hands of a clock be at right angles?

At what time between 4:00 and 5:00 will the hands of a clock be at right angles?

Step 1: Understand that a clock has two hands: the hour hand and the minute hand.
Step 2: Know that the hour hand moves 30 degrees for each hour (360 degrees / 12 hours).
Step 3: Calculate the position of the hour hand at 4:00. It is at 120 degrees (4 hours * 30 degrees).
Step 4: The minute hand moves 6 degrees for each minute (360 degrees / 60 minutes).
Step 5: Set up an equation to find when the angle between the two hands is 90 degrees.
Step 6: The angle between the hour hand and minute hand can be calculated as: |(30*hour - (11/2)*minutes)|.
Step 7: Substitute hour = 4 into the equation: |(120 - (11/2)*minutes)| = 90.
Step 8: Solve the equation: 120 - (11/2)*minutes = 90 or 120 - (11/2)*minutes = -90.
Step 9: For the first equation, solve for minutes: (11/2)*minutes = 30, so minutes = 60/11, which is approximately 5.45 minutes.
Step 10: For the second equation, solve for minutes: (11/2)*minutes = 210, so minutes = 420/11, which is approximately 38.18 minutes.
Step 11: The two times when the hands are at right angles are approximately 4:05 and 4:38.
Step 12: Since we are looking for the time between 4:00 and 5:00, the answer is approximately 4:20.

Clock Angles – Understanding the positions of the hour and minute hands on a clock and how to calculate the angles between them.
Time Calculation – Calculating the specific time when the hands of the clock form a right angle.