At what time after 12:00 o’clock midnight will the minute
hand and the hour hand of a clock be on a straight line for
the first time?
Since the Complete Angle is "360^{\\circ}"; therefore, the minute hand travels "360^{\\circ}"in one hour.
The clock is divided into "12" parts (hour); therefore, the hour hand travels "\\frac{360^{\\circ}}{12}=30^{\\circ}" in one hour.
Let "x" be the position of the minute hand on the clock (in minutes)
Since one hour has "60" minutes; therefore, one hour can be expressed as "\\frac{x}{60}".
So to find the position of the minute hand on the clock substitute into the relation:
Min hand in degrees - Hour hand in degrees = "180^{\\circ}"
"\\frac{x}{60}\\cdot 360-\\frac{x}{60}\\cdot 30=180^{\\circ}"
Simplify each fraction:
"6x-\\frac{1}{2}x=180"
Subtract the terms:
"5.5x=180"
Divide both sides by "5.5" :
"x\\approx 32.73"
So, at "12:32:73" both pm, the hands of a clock will be in a straight line.
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