Problem 56 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

At what time between 3 P.M. and 4 P.M. will the minute hand coincide with the hour hand? [Hint: The hour hand moves \(\frac{1}{12}\) as fast as the minute hand.]

Step-by-Step Solution

Answer

The minute and hour hands coincide at approximately 3:16:22 P.M.

Step 1: Define the Relationship between the Hands

Define the angular speeds of the hour and minute hands. The minute hand completes one full circle (360 degrees) in 60 minutes, so its speed is 6 degrees per minute. The hour hand completes one full circle in 12 hours (720 minutes), so its speed is 0.5 degrees per minute.

Step 2: Express the Positions of the Hands at 3 P.M.

At 3 P.M., the hour hand is at 90 degrees (since 360 degrees / 12 hours * 3 hours = 90 degrees). The minute hand is at 0 degrees.

Step 3: Set Up the Equation for When the Hands Coincide

Let t be the time in minutes after 3 P.M. The position of the hour hand after t minutes can be expressed as: 90 + 0.5t. The position of the minute hand after t minutes can be expressed as: 6t. Set the positions equal to one another to find t: 90 + 0.5t = 6t.

Step 4: Solve the Equation

Subtract 0.5t from both sides: 90 = 5.5t. Solve for t: t = 90 / 5.5 = 16.36 minutes.

Step 5: Convert Time to Standard Format

Convert 16.36 minutes to the appropriate time format. 16.36 minutes after 3 P.M. is approximately 3:16:22 P.M.

Key Concepts

Angular Speed

Angular speed refers to how fast an object rotates or revolves relative to another point, here typically the center of a clock. In the context of a clock, angular speed helps us understand how quickly the hour and minute hands move around the clock face.
For example:

• The minute hand completes a full circle (360 degrees) in 60 minutes. This means its angular speed is \( \frac{360}{60} = 6 \) degrees per minute.
• The hour hand, on the other hand, moves slower. It completes a full circle in 12 hours (or 720 minutes). Hence, its angular speed is \( \frac{360}{720} = 0.5 \) degrees per minute.

Understanding these speeds allows us to set up equations and solve for the moments they coincide.

Position of Hands

The position of the clock hands is crucial for solving clock problems. It is measured in degrees from the 12 o'clock position.
Let's break it down:

• At exactly 3 P.M., the hour hand is at 90 degrees because three hours represent one-quarter of the total 12 hours on the clock.
• The minute hand starts at 0 degrees at the top of the hour (12 o'clock position) and moves 6 degrees per minute.

By understanding the initial positions and how they change over time, we set up the stage to find when the hands coincide.

Solving Equations

To determine when the minute and hour hands overlap, we can create and solve an equation using their positions and angular speeds.
Consider this:

• Let \( t \) represent the number of minutes after 3 P.M. The position of the hour hand after \( t \) minutes is given by \( 90 + 0.5t \).
• The position of the minute hand after \( t \) minutes is \( 6t \).

We set these two expressions equal to each other to find when the hands align: \( 90 + 0.5t = 6t \).
Solving for \( t \) involves:

• Subtracting \( 0.5t \) from both sides yields \( 90 = 5.5t \).
• Dividing both sides by 5.5 gives \( t \approx 16.36 \) minutes.

This means the hands coincide approximately 16.36 minutes after 3 P.M.

Time Conversion

Once we solve the equation, we need to convert the time into a standard format to understand it better.
Here's how:

• From our previous calculation, we found \( t \approx 16.36 \) minutes. This needs conversion to the standard clock time.
• 16.36 minutes can be split into 16 minutes and 0.36 of a minute. Since there are 60 seconds in a minute, \( 0.36 \times 60 = 21.6 \) seconds.

Thus, the final time when the hands coincide would be 3:16:22 P.M. approximately.
Recognizing and correctly performing this time conversion ensures accurate results when interpreting solutions.