Problem 98 - Precalculus : Building Concepts and Connections

Question

A scamstress secures one end of a piece of thread to a spool. Then she uses an attachment on her scwing machine to cause the spool to spin around, which in turn causes the thread to wind around the spool. If the spool has a diameter of \(1.6 \mathrm{cm}\) and spins at a rate of 3 revolutions per second, what length of thread (in centimeters) is wound around the spool in 1 second?

Step-by-Step Solution

Answer

To find out the length of the thread wound around the spool in one second, the circumference of the spool due to the diameter is firstly calculated. Then the length of the thread used per revolution which is equal to the circumference is determined. Lastly, the total length of the thread used in one second is calculated by multiplying the spool's rate of revolution per second by the length of the thread used per revolution. Substitute the given diameter into the formula and use 3.14 as the value of \(\pi\) to calculate the numerical solution.

Step 1: Calculate the Circumference of the Spool

Firstly, let's calculate the circumference of the spool using the formula \(C = \pi d\). Here, the diameter (\(d\)) given is 1.6 cm. Hence, \(C = \pi \times 1.6\).

Step 2: Determine the length of the thread per revolution

The length of the thread used in a single revolution around the spool is equal to the circumference of the spool. Which we have calculated in the previous step.

Step 3: Calculate the Total Length of the Thread in One Second

As given, the spool spins at a rate of 3 revolutions per second. Therefore, in one second, the length of the thread used is \(3 \times C\), where \(C\) is the circumference of the spool.

Key Concepts

Circumference Calculation

When you want to find out how far something goes around a circular object, like a spool, you need to calculate its circumference. Circumference is the perimeter or boundary line of a circle. To find this, use the formula for the circumference, which is

• Formula: \( C = \pi d \)
  
• Here, \( d \) is the diameter of the circle.

In our example, the diameter of the spool is given as 1.6 cm.
So, we plug this value into the formula to get:

• Circumference: \( C = \pi \times 1.6 \)

Understanding how to find the circumference is essential for calculating the length of thread wound around the spool.

Revolution Rate

The revolution rate is simply how fast something spins. It tells us how many times an object, like our spool, rotates completely in a specific period, such as a second.
In this context, knowing the revolution rate is crucial because it helps us understand how much thread wraps around the spool with each complete turn.

• Rate Given: The spool spins at 3 revolutions per second.
• This means in one full second, the spool would make 3 complete turns.

This information plays a key role in determining how much thread gets used in a given time frame.
Making note of the revolution rate is an excellent way to predict and calculate continuous motion.

Thread Length Calculation

Now that we have our key pieces of information—the circumference and the revolution rate—combining these allows us to calculate the length of thread wrapped around the spool. The task here is to find this length over 1 second.
Using the circumference from our earlier calculation, we consider the revolution rate:

• Each Revolution uses: Thread Length equivalent to the Circumference.
• Total Thread Length in 1s: \( 3 \times C \)

Since the spool spins 3 times in one second, we simply multiply the circumference by 3.
This gives us the total length of thread wound around the spool in one second, which helps in understanding both theoretical and practical calculations in real-world tasks.