In 1 h the minute hand on a clock moves through a complete circle, and the hour hand moves through 1 12 of a circle. Through how many radians do the minute hand and the hour hand move between 1:00 p.m. and 1:45 p.m. (on the same day)?

The question given says that the minute hand on a clock moves through complete circle in 1 hour, that is 360° or 2π. It also says that the hour hand moves through 1/12 of a circle, that means 30° or π/6.

To know how many radians do the minute hand and the hour hand move between 1:00 p.m. and 1:45 p.m, it's necessary to calculate how many radians move them per minute.

Between 1:00 p.m. and 1:45 p.m 45 minutes have passed. With that information, the radians can be calculated using multiplication and division.

Minute hand: To know how many radians move the minute hand per minute division wil be used.

Movement in an hour/ minutes in an hour

2π rad/60 min= π/30 rad-min

That means the minute hand move π/30 radians in a minute.

Now, multiplication can be used to calculate how many radians move the minute hand in 1h.

(π/30 rad-min)(45 minutes)= 3/2π rad

The minute hand moves 3/2π radians between 1:00 p.m. and 1:45 p.m.

Hour hand: To know how many radians move the hour hand per minute division wil be used.

Movement in an hour/ minutes in an hour

2π rad/(60 min x 12 hours)= π/360 rad-min

That means the minute hand move π/360 radians in a minute.

Now, multiplication can be used to calculate how many radians move the hour hand in 1h.

(π/360 rad-min)(45 minutes)= π/8 rad

The minute hand moves π/8 radians between 1:00 p.m. and 1:45 p.m.