Clock Angle Calculation at 2:30
To find the angle between the hour and minute hands of a clock, we use a standard formula. The calculation considers the specific positions of both hands at the given time.
Angle Formula for Clock Hands
The angle $|\theta|$ between the hour hand (H) and the minute hand (M) can be calculated using the formula:
$|\theta| = |30H - 5.5M|$
Where:
- $H$ is the hour value (for 2:30, H = 2).
- $M$ is the minute value (for 2:30, M = 30).
Calculating the Angle at 2:30
Substitute H = 2 and M = 30 into the formula:
$|\theta| = |(30 \times 2) - (5.5 \times 30)|$
Perform the multiplications:
$|\theta| = |60 - 165|$
Calculate the difference:
$|\theta| = |-105|$
Take the absolute value:
$|\theta| = 105^\circ$
Identifying the Obtuse Angle
The calculated angle is $105^\circ$. The question asks for the obtuse angle. An obtuse angle is greater than $90^\circ$ and less than $180^\circ$. Since $105^\circ$ fits this definition, it is the required obtuse angle.
The reflex angle, $360^\circ - 105^\circ = 255^\circ$, is not the obtuse angle requested.
Final Result
The obtuse angle formed by the clock hands at 2:30 is $105^\circ$.
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