Question

consider a simple clock.
a. find the measure of the angle swept by the hour hand between noon and 6:45 p.m.
b. find the exact angle between the minute and hour hands at 1:25 p.m.
a. the measure of the angle is □°□

Explanation:

Step1: Calculate angle per hour for hour - hand

The hour - hand of a clock completes a full rotation ($360^{\circ}$) in 12 hours. So the angle it sweeps per hour is $\frac{360^{\circ}}{12}=30^{\circ}$ per hour.

Step2: Calculate angle per minute for hour - hand

Since there are 60 minutes in an hour, the angle the hour - hand sweeps per minute is $\frac{30^{\circ}}{60}=0.5^{\circ}$ per minute.

Step3: Calculate angle swept by hour - hand from noon to 6:45 PM

From noon to 6:00 PM, 6 hours have passed, and the angle swept is $6\times30^{\circ}=180^{\circ}$. From 6:00 PM to 6:45 PM, 45 minutes have passed, and the angle swept in these 45 minutes is $45\times0.5^{\circ}=22.5^{\circ}$. So the total angle swept is $180^{\circ}+22.5^{\circ}=202.5^{\circ}=202^{\circ}30'$.

Step4: Calculate position of hour - hand at 1:25 PM

At 1:00 PM, the hour - hand is at an angle of $1\times30^{\circ}=30^{\circ}$ from the 12 - o'clock position. In 25 minutes, the hour - hand moves an additional $25\times0.5^{\circ}=12.5^{\circ}$. So the position of the hour - hand at 1:25 PM is $30^{\circ}+12.5^{\circ}=42.5^{\circ}$ from the 12 - o'clock position.

Step5: Calculate position of minute - hand at 1:25 PM

The minute - hand completes a full rotation ($360^{\circ}$) in 60 minutes. So at 1:25 PM, the minute - hand is at an angle of $\frac{25}{60}\times360^{\circ}=150^{\circ}$ from the 12 - o'clock position.

Step6: Calculate angle between minute and hour hands at 1:25 PM

The angle between the minute and hour hands at 1:25 PM is $150^{\circ}-42.5^{\circ}=107.5^{\circ}=107^{\circ}30'$.

Answer:

a. $202^{\circ}30'$
b. $107^{\circ}30'$