Consider a correctly functioning clock that starts ticking at noon and find the time between noon and

Write an expression involving the positions of the minute and hour hands after m minutes. Set that expression equal to the given angle measure and solve for m.

Begin by expressing the position of the hour hand after m minutes. In 12​ hours, the hour hand moves through a full​ circle, or

360 degrees.  Use this fact to find the number of degrees the hour hand will move in one hour. The hour hand moves 30degrees in one hour.  Divide to find the number of degrees the hour hand moves in one minute.  30 degrees divided by 60equals0.5degrees

The angle of the hour hand after m minutes is left parenthesis 0.5 m right parenthesis degrees.

Now express the position of the minute hand after m minutes. In one​ hour, the minute hand moves through a full​ circle, or

360 degrees. Use this fact to find the number of degrees the minute hand will move in one minute.

The minute hand moves  6 degrees in one minute. So the angle of the minute hand after m minutes is ​(6m) degrees.

The angle between the hour hand and the minute hand after m minutes is given by the expression on the left side of the equation below. The right side of the equation is the given angle measure.

105

105

Solve for​ m, rounding to three decimal places.

equals

19.091

​Therefore, the angle between the minute hand and the hour hand will be

105

degreesafter about

19.091

minutes. Because there are 60 seconds in a​ minute, to find the number of​ seconds, multiply the decimal remainder by​ 60, rounding to the nearest second.

0.091

times 60equals5

​Therefore, the angle between the minute hand and the hour hand will be

105

degreesat

12

​:19​:05P.M.