Understanding Radian Measures and Clock Angles

Problem 7: Radian Measure for String Collage

• An art student aims to create a string collage that connects six equally spaced points on the circumference of a circle to the circle's center.
• To determine the radian measure of the angle between two adjacent pieces of string:
  • A circle has a total angular measure of $2\pi$ radians.
  • For six points, the total angle must be divided by the number of connections (which is equivalent to the number of points):
  • Total angles between the points = $\frac{2\pi}{6}$ radians.
  • Simplifying this gives:
  • $\frac{2\pi}{6} = \frac{\pi}{3}$ radians.

Problem 8: Angle Between Clock Hands at 2:00 p.m.

• To find the radian measure of the angle formed by the hands of a clock at 2:00 p.m.:
  • The hour hand is on 2, approximately $\frac{2}{12}$ of the way around the clock face.
  • The minute hand points at 12 (0 degrees or 0 radians).
  • The angle in degrees between the hour hand at 2 and the minute hand at 12 is:
  • $2 \times 30 = 60$ degrees (since each hour represents 30 degrees).
  • Convert degrees to radians:
  • To convert degrees to radians, use the conversion factor: $\frac{\pi}{180}$.
  • Thus, the angle in radians is:
    • $\frac{60\pi}{180} = \frac{\pi}{3}$ radians.

Problem 9: Angle Between Clock Hands at 5:00 p.m.

• To calculate the angle formed by the hands of a clock at 5:00 p.m.:
  • The hour hand indicates 5:
  • In degrees, the angle is: $5 \times 30 = 150$ degrees.
  • The minute hand is at 12 (0 degrees or 0 radians).
  • Convert the angle to radians:
  • This gives: $\frac{150\pi}{180} = \frac{5\pi}{6}$ radians.

Problem 10: Minute Hand Movement in 24 Minutes

• To determine how many radians the minute hand of a clock turns in 24 minutes:
  • The minute hand completes a full circle (360 degrees or $2\pi$ radians) in 60 minutes.
  • In 1 minute, it moves:
  • $\frac{2\pi}{60}$ radians.
  • Therefore, in 24 minutes, it will turn:
  • $\frac{2\pi}{60} \times 24 = \frac{48\pi}{60} = \frac{4\pi}{5}$ radians.

Problem 11: Minute Hand Movement in 35 Minutes

• To find the radians turned by the minute hand in 35 minutes:
  • In 1 minute, the movement is still: $\frac{2\pi}{60}$ radians.
  • Thus, for 35 minutes:
  • $\frac{2\pi}{60} \times 35 = \frac{70\pi}{60} = \frac{7\pi}{6}$ radians.

Problem 12: Minute Hand Movement in 100 Minutes

• To find the number of radians turned by the minute hand in 100 minutes:
  • Again, using the same one-minute conversion: $\frac{2\pi}{60}$ radians.
  • For 100 minutes:
  • $\frac{2\pi}{60} \times 100 = \frac{200\pi}{60} = \frac{10\pi}{3}$ radians.