Question

What is the closest time between 7 and 8 when the hands of your watch are exactly opposite each other?

• A. 7 Hr - 5 Min
• B. 7 Hr - 5.5 Min
• C. 7 Hr - 6 Min
• D. 7 Hr - 6.5 Min

Answer 1

The Correct Option is B

The problem we need to solve is determining the exact time between 7 and 8 o'clock when the hands of a watch are diametrically opposite each other. In this scenario, the hour and minute hands form a 180-degree angle.

We will calculate this using the formula for the angle between the minute and hour hands of a clock: Angle = |(30*hour - 11/2*minutes)|. We set the equation to find the time when this angle is 180 degrees.

1. Let the time be 7 hr and x min.

2. The position of the hour hand from 12 o'clock is (30*7 + 30/60*x) = (210 + 0.5x) degrees.

3. The position of the minute hand from 12 o'clock is (6*x) degrees.

Equation: |(210 + 0.5x) - 6x| = 180.

4. Simplify: |210 + 0.5x - 6x| = 180.

5. Combine terms: |210 - 5.5x| = 180.

6. Remove the absolute value by considering two equations:
(i) 210 - 5.5x = 180
(ii) 210 - 5.5x = -180

7. Solve (i):
210 - 5.5x = 180
5.5x = 30
x = 30/5.5 ≈ 5.4545

8. Solve (ii):
210 - 5.5x = -180
5.5x = 390
x = 390/5.5 ≈ 70.9091 (invalid since it's beyond 60 minutes)

Therefore, the valid solution is approximately 7 hr 5.5 min.

The closest time when the watch hands are exactly opposite each other between 7 and 8 is 7 Hr - 5.5 Min.