Problem 88 - Algebra and Trigonometry

Question

Clocks and Angles In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through \(\frac{1}{12}\) of a circle. Through how many radians do the minute and the hour hand move between \(1 : 00 \mathrm{P.M.}\) and \(6 : 45\) P.M. (on the same day)?

Step-by-Step Solution

Answer

Minute hand: \(11.5\pi\) radians, Hour hand: \(\frac{5.75\pi}{6}\) radians.

Step 1: Calculating the Angle for the Minute Hand per Hour

The minute hand moves through a full circle which is 360 degrees in an hour. Since there are \(2\pi\) radians in a full circle, the minute hand moves \(2\pi\) radians each hour. We need to determine how many hours and minutes the minute hand moves from 1:00 P.M. to 6:45 P.M.

Step 2: Calculating the Time Duration

From 1:00 P.M. to 6:45 P.M. is 5 hours and 45 minutes. In decimal form, this is 5.75 hours because 45 minutes is \(\frac{45}{60} = 0.75\) hours.

Step 3: Calculating the Total Angle for the Minute Hand

The minute hand moves \(2\pi\) radians in one hour. In 5.75 hours, it moves (\(2\pi \times 5.75\)) radians. Therefore, the total angle the minute hand moves is \(11.5\pi\) radians.

Step 4: Calculating the Angle for the Hour Hand per Hour

The hour hand moves through \(\frac{1}{12}\) of a circle each hour, which is \(\frac{2\pi}{12} = \frac{\pi}{6}\) radians per hour.

Step 5: Calculating the Total Angle for the Hour Hand

The hour hand moves \(\frac{\pi}{6}\) radians in one hour. In 5.75 hours, it moves (\(\frac{\pi}{6} \times 5.75\)) radians. Therefore, the total angle the hour hand moves is \(\frac{5.75\pi}{6}\) radians.

Step 6: Final Results

The minute hand moves \(11.5\pi\) radians, and the hour hand moves \(\frac{5.75\pi}{6}\) radians in 5 hours and 45 minutes (from 1:00 P.M. to 6:45 P.M.).

Key Concepts

Radian Measure

The radian is a way to measure angles, and it's incredibly useful in trigonometry. Rather than using degrees, which splits a circle into 360 slices, radians are based on the radius of the circle itself. A full circle is equal to \(2\pi\) radians. This is because the circumference of a circle, which is the full rotation, is \(2\pi\) times the radius.

• 1 radian is roughly 57.3 degrees.
• \(\pi\) radians equals 180 degrees, which is a half-circle.
• This measure connects arc length to the radius, making calculations simpler when dealing with circular motions such as those of clock hands.

Knowing how to convert degrees into radians and vice versa is crucial. For instance, if you know the degrees, you can convert to radians by multiplying the degrees by \(\frac{\pi}{180}\). Conversely, converting from radians to degrees involves multiplying by \(\frac{180}{\pi}\). These conversions are essential when understanding how much a clock's hand moves in terms of radians.

Angles

Angles are a fundamental part of geometry and trigonometry. An angle measures how far one line is rotated from another line. In clocks, we use angles to determine how far the hands of the clock have turned.

• A full circle is 360 degrees or \(2\pi\) radians.
• Adjusting to half-circles means 180 degrees or \(\pi\) radians.
• A quarter circle measures 90 degrees or \(\frac{\pi}{2}\) radians.

Each position of the clock's hands points to a different angle. The minute hand of a clock, for example, completes a full circle in one hour, marking \(2\pi\) radians. Meanwhile, the hour hand advances by \(\frac{\pi}{6}\) radians every hour. Understanding these angles helps you calculate how far the hands move as time passes on the clock.

Clock Problems

Clock problems often involve calculating the movement of clock hands in terms of angles or radians. These problems may require you to determine how many degrees or radians the hands turn over a given period. Here's a basic rundown:

• The minute hand completes a \(2\pi\) radian or 360-degree rotation every hour.
• The hour hand shifts \(\frac{\pi}{6}\) radians or 30 degrees per hour.

To tackle these problems, first calculate the total time interval you're interested in, such as from 1:00 P.M. to 6:45 P.M. Next, convert this time into decimal form for ease of calculation (e.g., 5 hours and 45 minutes becomes 5.75 hours). Then, apply the rate of movement to find out how many radians each hand covers during that time. Solving clock problems sharpens your skills in both radian measure and angle calculation, allowing you to understand and visualize the passage of time in mathematical terms.