Let's first consider at what time(s) during the day the hands of a clock form a 48-degree angle.
The angle formed by the hands of a clock is given by the formula:
θ = |(30h - 11/2)m|
where:
- h is the hour hand position (measured in hours from the 12 o'clock position)
- m is the minute hand position (measured in minutes from the 12 o'clock position)
Since we want to find at what other times during the day the same hands on the clock form a 48-degree angle, we can set this equation equal to 48 degrees and solve for other values of m and h.
48 = |(30h - 11/2)m|
Since the absolute value of a product is equal to the product of the absolute values, we can split this equation into two cases:
Case 1: (30h - 11/2)m = 48
30h - 11/2 = 1 (because m cannot be negative)
h = 13/15
This means that the hands of the clock are at the same angle at 1:13 and approximately 43 seconds (rounded to the nearest second).
Case 2: (30h - 11/2)m = -48
30h - 11/2 = -1 (because m cannot be negative)
h = 23/15
This means that the hands of the clock are at the same angle at 11:23 and approximately 38 seconds (rounded to the nearest second).
Therefore, the same hands on the clock form a 48-degree angle at 1:13 and approximately 43 seconds and at 11:23 and approximately 38 seconds.