Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Find the degree measure of the smaller angle formed by the hands of a clock at the following times. Do not use a calculator. $$3: 15$$

Step-by-Step Solution

Answer

The angle is 7.5 degrees.

Step 1: Understand Clock Positioning

A clock has 12 hours, and full rotation (360 degrees) is divided into 12 parts. Thus, each hour corresponds to \( \frac{360}{12} = 30 \) degrees. Additionally, each minute corresponds to \( \frac{1}{60} \times 30 = 0.5 \) degrees for the hour hand.

Step 2: Calculate the Angle of the Hour Hand

At 3:00, the hour hand is exactly on the 3, which is \( 3 \times 30 = 90 \) degrees from 12 o'clock. At 3:15, the hour hand has moved further by \( 15 \times 0.5 = 7.5 \) degrees. So, the hour hand is at \( 90 + 7.5 = 97.5 \) degrees.

Step 3: Calculate the Angle of the Minute Hand

The minute hand at 15 minutes is pointed exactly at the 3, which corresponds to \( 15 \times 6 = 90 \) degrees, since each minute represents 6 degrees (\( \frac{360}{60} \)).

Step 4: Calculate the Difference in Angles

The angle between the hour hand and the minute hand is the absolute difference between their angles. So, the angle formed is \( |97.5 - 90| = 7.5 \) degrees.

Key Concepts

Angle Measurement

When calculating angles on a clock face, it's important to understand how the circle is divided. A complete circle has 360 degrees. On a clock face, these 360 degrees are divided by the 12 hours, which means that each hour marker on the clock represents a segment of \( 30 \) degrees

• This division makes it easy to determine the angle for each hour position. For example, the 3 o'clock position is \( 3 \times 30 = 90 \) degrees from the 12 o'clock starting point.
• For minute calculations, since there are 60 minutes in an hour, each minute accounts for \( \frac{360}{60} = 6 \) degrees.

When looking at angles, it's also important to consider the direction of measurement; angles are typically measured in a clockwise direction from the 12 o'clock position.

Hour Hand Movement

The hour hand of a clock moves gradually as time progresses. Unlike the minute hand, which jumps from one numeral to the next, the hour hand moves continuously. This movement means we need to consider its position closely, even when it's still in between the numbers.

• At the start of each hour, the hour hand is perfectly aligned with the number on the clock. For example, at 3:00, it's at 90 degrees.
• As time progresses past the hour, the hour hand continues to move. This movement can be calculated by recognizing that each minute passed adds \( 0.5 \) degrees to the hour hand's position, given that 30 degrees are spanned by 60 minutes.

So, at 3:15, the hour hand has moved an additional \( 15 \times 0.5 = 7.5 \) degrees past the 90-degree mark.

Minute Hand Position

The minute hand moves at a steady pace, with each minute contributing to its position on the clock face.

• For every minute that passes, the minute hand moves 6 degrees. This is because 360 degrees divided by 60 minutes equals 6 degrees per minute.
• Importantly, the minute hand's position can also coincide with hour markers, making calculation straightforward at quarter-hour marks. For instance, at 15 minutes past the hour, the minute hand points to the 3 which is exactly 90 degrees.

Understanding the movement of the minute hand allows us to see where it aligns on a clock face at any given time. This can help us easily determine it relative to the hour hand.