If a clock shows 8:00, at what time will the hands be opposite each other for the first time?

Steps to Solve

1. Calculate the initial positions of the hour and minute hands at 8:00

At 8:00, the hour hand is at 240 degrees (8 hours multiplied by 30 degrees per hour) and the minute hand is at 0 degrees.

2. Establish the rate of movement for both hands

The hour hand moves at a rate of $0.5$ degrees per minute (30 degrees per hour / 60 minutes) and the minute hand moves at $6$ degrees per minute (360 degrees per hour / 60 minutes).

3. Set up the equation for when the hands are 180 degrees apart

The positions of the hour and minute hands, $H(t)$ and $M(t)$ respectively, can be expressed as:

• $H(t) = 240 + 0.5t$
• $M(t) = 6t$

We can set up the equation for when they are 180 degrees apart: $$ |H(t) - M(t)| = 180 $$

4. Solve for $t$ using the equation

We will solve two cases:

• Case 1: $H(t) - M(t) = 180$ $$ 240 + 0.5t - 6t = 180 $$ $$ 240 - 180 = 5.5t $$ $$ 60 = 5.5t $$ $$ t = \frac{60}{5.5} \approx 10.91 $$

• Case 2: $M(t) - H(t) = 180$ $$ 6t - (240 + 0.5t) = 180 $$ $$ 6t - 240 - 0.5t = 180 $$ $$ 5.5t = 420 $$ $$ t = \frac{420}{5.5} \approx 76.36 $$

5. Determine the valid time after 8:00

Now we determine the valid times from both cases:

• From Case 1:

  • $t \approx 10.91$ minutes => approximately 8:11 (and a few seconds)

• From Case 2:

  • $t \approx 76.36$ minutes => 1 hour and 16.36 minutes after 8:00 => approximately 9:16 (and a few seconds)

The first valid time is approximately 8:11.