At what time between 8 oclock and 9 oclock will the hands of a clock be in the same straight line but not coincide with each other?

Introduction / Context:
This problem asks us to find the time between 8 and 9 oclock when the hands lie in the same straight line but do not coincide. That means they are 180 degrees apart. It is similar in structure to other straight line questions, but the starting hour and resulting fraction differ. The method involves setting up the angular positions of both hands and solving for the time in minutes after 8 oclock.

Given Data / Assumptions:

• Time interval is between 8:00 and 9:00.
• Hands are in the same straight line but not together, so the angle between them is 180 degrees.
• Minute hand angle = 6 degrees per minute.
• Hour hand angle = 30 degrees per hour plus 0.5 degree per minute, starting from 240 degrees at 8:00.

Concept / Approach:
Let t be the number of minutes past 8:00. Then: hour hand angle = 30 * 8 + 0.5 * t = 240 + 0.5 t, minute hand angle = 6 * t. We require the angle between them to be 180 degrees, so: |6 t - (240 + 0.5 t)| = 180. We solve this equation and choose the solution with t between 0 and 60 minutes.

Step-by-Step Solution:
Step 1: Set up the equation. |6 t - (240 + 0.5 t)| = 180. There are two possibilities, but only one will fall between 0 and 60 minutes. Case 1: 6 t - (240 + 0.5 t) = 180. 5.5 t - 240 = 180. 5.5 t = 420. t = 420 / 5.5 = 840 / 11 = 76 4/11 minutes, which is more than 60 minutes and therefore outside the hour from 8 to 9. Case 2: (240 + 0.5 t) - 6 t = 180. 240 - 5.5 t = 180. 5.5 t = 60. t = 60 / 5.5 = 120 / 11 minutes. 120 / 11 is approximately 10 10/11 minutes, which lies between 0 and 60 minutes. Thus, the time is 120 / 11 minutes past 8.

Verification / Alternative check:
At t = 120 / 11: Minute hand angle = 6 * 120 / 11 = 720 / 11 degrees. Hour hand angle = 240 + 0.5 * 120 / 11 = 240 + 60 / 11 = (2640 + 60) / 11 = 2700 / 11 degrees. Difference = (2700 / 11) - (720 / 11) = 1980 / 11 = 180 degrees, confirming the hands are in a straight line but opposite each other.

Why Other Options Are Wrong:
Options a, c, and d (100/11, 90/11, 80/11 minutes) do not yield exactly 180 degrees of separation when substituted into the angle formulas. They correspond to other separations that are either acute or obtuse but not a straight line.

Common Pitfalls:
A common mistake is to accept the first solution t = 76 4/11 minutes without checking that it exceeds 60 minutes and therefore does not fall in the specified hour. Another is forgetting that the hour hand also moves, leading to equations that use 240 degrees as a fixed value and ignore the 0.5 t term.

Final Answer:
The hands of the clock are in the same straight line but not together at 120/11 minutes past 8.