Problem 67 - Precalculus

Question

Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in 20 minutes?

Step-by-Step Solution

Answer

The minute hand traces 120 degrees in 20 minutes.

Step 1: Understand the Full Rotation of the Minute Hand

The minute hand completes one full rotation (360 degrees) in 60 minutes. This means that in 60 minutes, the minute hand traces a total angle of 360 degrees.

Step 2: Find the Angle Per Minute

Since 360 degrees are completed in 60 minutes, we calculate the angle moved per minute by dividing 360 degrees by 60 minutes:\[\text{Angle per minute} = \frac{360}{60} = 6\text{ degrees per minute}\]

Step 3: Calculate the Angle for 20 Minutes

To find the angle traced by the minute hand in 20 minutes, multiply the angle moved per minute by 20:\[\text{Angle for 20 minutes} = 6 \times 20 = 120\text{ degrees}\]

Key Concepts

Minute Hand

When it comes to telling time on a clock, the minute hand plays a crucial role in tracking the passing minutes. The minute hand is the longer of the two hands found on traditional analog clocks.
It completes a full circle as the hour progresses.

• One full rotation equals 360 degrees.
• Takes exactly 60 minutes to complete this full rotation.

This means that each minute contributes to a small part of the movement around the clock face. Understanding this concept is essential when calculating angles traced by the minute hand over a certain period. Breaking down how much the minute hand moves helps in accurately determining the angles involved in various time intervals.

Angle Calculation

Angle calculation in terms of the minute hand involves figuring out how far the minute hand travels within a given timeframe. Knowing that the entire clock is a circle of 360 degrees, we can determine how much movement occurs per individual minute.The key steps include:

• Realizing that in 60 minutes, the minute hand covers the whole circle.
• Determining the angle move per single minute by dividing 360 degrees by 60.

The math is simple but critical for understanding:\[\text{Angle per minute} = \frac{360}{60} = 6 \text{ degrees}\]This calculation provides a straightforward way to determine how much the minute hand moves in any set amount of minutes. For example, if you need to find out the angle in 20 minutes, multiply the degrees per minute by the given time:\[\text{Angle for 20 minutes} = 6 \times 20 = 120 \text{ degrees}\] This comprehensive method allows you to find the angle traced in both short and extended time frames.

Rotation Degree

Rotation degree is a term used to describe the circle created as the clock's hand moves over time. This is vital for visualizing and comprehending how angles construct on a clock face, merging both time and geometry. Here's a breakdown of what to consider:

• A full rotation equals 360 degrees, which corresponds to one complete cycle of the minute hand.
• Calculating smaller segments of this rotation helps understand time intervals.

For smaller intervals, it's important to grasp that the minute hand rotates through known angles, like so: - When moving for 1 minute, it rotates 6 degrees. - For arbitrary minutes, multiply the single-minute angle by the number of minutes. The concept of rotation degree lets you convert time into angular measurement, a useful skill for solving clock angle problems. It's the bridge between understanding time passage and its geometric representation on the clock face.