Consider the motion of the hour hand of a clock. What angle does the hour hand make with respect to the vertical 12 o'clock position when it is at the times below? Express each angle in both degrees and radians, and measure angles going clockwise from the vertical axis.

5 o clock?

Five O'clock occurs when the hour hand is 5/12 of 360 degrees past the vertical minute hand.

5/12 x 360 = 150 degrees

Convert that to radians for the other answer.

60

To determine the angle the hour hand makes with respect to the vertical 12 o'clock position at 5 o'clock, we need to calculate the hour hand's position relative to the vertical axis.

The hour hand completes a full revolution (360°) or 2π radians in 12 hours.

At 5 o'clock, the hour hand is at a quarter of the way between 4 and 5. This corresponds to a time fraction of 5/12.

To calculate the angle in degrees, multiply the time fraction by 360°:

(5/12) * 360° = 150°

To calculate the angle in radians, multiply the time fraction by 2π:

(5/12) * 2π = (5/6)π radians

Therefore, the hour hand makes an angle of 150° or (5/6)π radians with respect to the vertical 12 o'clock position at 5 o'clock.

To determine the angle made by the hour hand with respect to the vertical 12 o'clock position, we need to understand the motion of the hour hand and the relationship between time and angle.

The hour hand of a clock completes one full revolution (360 degrees) in 12 hours. So, the hour hand moves 360 degrees in 12 hours, which is equivalent to 30 degrees per hour.

To find the angle made by the hour hand at 5 o'clock, we need to determine the elapsed time since 12 o'clock and multiply it by the rate of motion (30 degrees per hour).

The time elapsed from 12 o'clock to 5 o'clock is 5 hours. Multiplying this by the rate of motion (30 degrees per hour), we get:

Angle = 5 hours × 30 degrees/hour
= 150 degrees

To convert this angle to radians, we can use the conversion factor: 1 radian = 180 degrees / π radians. Therefore:

Angle (in radians) = 150 degrees × (π radians / 180 degrees)
= (5π) / 6 radians

So, the angle made by the hour hand at 5 o'clock is 150 degrees or (5π) / 6 radians, measured clockwise from the vertical axis.