At what time between 4 oclock and 5 oclock will the hands of a clock be at a right angle to each other?

Introduction / Context:
This question asks for a time between 4 and 5 oclock when the hands of a clock are at a right angle, that is, 90 degrees apart. The method is similar to other clock angle problems where we express the angles of both hands as functions of the minutes past the hour and then enforce the condition on their difference.

Given Data / Assumptions:

• Time interval is from 4:00 to 5:00.
• Right angle means the angle between the hands is exactly 90 degrees.
• Minute hand moves 6 degrees per minute.
• Hour hand moves 30 degrees per hour plus 0.5 degree per minute.

Concept / Approach:
Let t be the number of minutes after 4:00. Then: hour hand angle = 30 * 4 + 0.5 * t = 120 + 0.5 t, minute hand angle = 6 * t. We need: |6 t - (120 + 0.5 t)| = 90. We solve this equation and choose the value of t between 0 and 60 minutes that fits.

Step-by-Step Solution:
Step 1: Form the equation. |6 t - (120 + 0.5 t)| = 90. This gives two possible equations: Case 1: 6 t - (120 + 0.5 t) = 90. Case 2: (120 + 0.5 t) - 6 t = 90. Step 2: Solve Case 1. 6 t - 120 - 0.5 t = 90. 5.5 t = 210. t = 210 / 5.5 = 420 / 11 minutes, which is about 38 2/11 minutes, within the hour but corresponds to another right angle time. Step 3: Solve Case 2. 120 + 0.5 t - 6 t = 90. 120 - 5.5 t = 90. 5.5 t = 30. t = 30 / 5.5 = 60 / 11 minutes. 60 / 11 = 5 5/11 minutes. Thus, one right angle time between 4 and 5 is 4:05 5/11. The option list refers to this first right angle instant.

Verification / Alternative check:
At t = 60 / 11: Hour hand angle = 120 + 0.5 * 60 / 11 = 120 + 30 / 11 = (1320 + 30) / 11 = 1350 / 11 degrees. Minute hand angle = 6 * 60 / 11 = 360 / 11 degrees. Difference = (1350 / 11) - (360 / 11) = 990 / 11 = 90 degrees, confirming the right angle condition.

Why Other Options Are Wrong:
Options a, b, and d (3 5/11, 4 5/11, 6 5/11 minutes past 4) do not yield an exact 90 degree separation when substituted into the formulas. The second right angle within the hour occurs later and does not match any of these choices, while 5 5/11 minutes past 4 matches the first right angle exactly and is given as option c.

Common Pitfalls:
A typical error is to forget that there are two right angle times in each hour except in special cases, and then mis identify which one matches the options. Another pitfall is solving only one case from the absolute value equation and missing the valid solution, or incorrectly approximating fractions and failing to recognise 5 5/11 minutes as the exact solution.

Final Answer:
The hands of the clock are at right angles at 5 5/11 minutes past 4.