Problem 52 - Basic College Mathematics

Question

The pitch of a screw is \(\frac{3}{32}\) in. How many complete rotations are necessary to drive the screw \(\frac{3}{4}\) in. into a piece of pine wood?

Step-by-Step Solution

Answer

24 rotations

Step 1: Understand pitch definition

Pitch is defined as the distance a screw moves forward with one complete rotation. Here, the pitch of the screw is given as \(\frac{3}{32}\) inches.

Step 2: Set up the ratio for the rotations

To find the number of rotations, divide the total distance that the screw needs to travel by the pitch: \( \frac{\text{Total distance}}{\text{Pitch}} \). In this case, you need to drive the screw \(\frac{3}{4}\) inches into the wood.

Step 3: Calculate the number of rotations

Use the provided values: \( \frac{3/4}{3/32} \). This simplifies as follows: \[ \frac{3 \times 32}{4 \times 3} = 24 \]. Therefore, 24 complete rotations are needed.

Key Concepts

Pitch Definition

The concept of pitch is crucial in understanding screw mechanics. Pitch refers to the distance a screw will travel forward with one complete turn. For example, if the pitch is \( \frac{3}{32} \) inches, it means that every full rotation of the screw moves it forward by \( \frac{3}{32} \) inches. Knowing the pitch helps in making accurate calculations for distance and rotations. It gives a clear idea of how many turns are required to achieve a specific depth.

Distance Calculation

Distance calculation is important when working with screws. In this exercise, you need to drive the screw \( \frac{3}{4} \) inches into a piece of wood. The pitch (distance per rotation) helps determine how many full rotations are needed to reach this depth. If the pitch is \( \frac{3}{32} \) inches, you divide the total distance by this pitch. This division will give you the number of rotations required for the screw to travel the specified distance.

Ratios and Proportions

Understanding ratios and proportions can simplify complex calculations. Here, the ratio helps us compare the needed distance with the pitch of the screw. If you have a total distance of \( \frac{3}{4} \) inches and a pitch of \( \frac{3}{32} \) inches, setting up a proportion \[ \frac{\text{Total distance}}{\text{Pitch}} = \frac{\frac{3}{4}}{\frac{3}{32}} \] aids in finding how many times the pitch (\( \frac{3}{32} \) inches per rotation) fits into the total distance. This is a straightforward way to understand the number of rotations.

Fraction Division

Fraction division plays a significant role in these calculations. To find the number of complete rotations required, we divide the total distance by the pitch. For instance, \( \frac{3}{4} \) divided by \( \frac{3}{32} \) can be written as \[ \frac{3/4}{3/32} = \frac{3}{4} \div \frac{3}{32} = \frac{3}{4} \times \frac{32}{3} \] By simplifying this, the complex fraction reduces to \[ \frac{3 \times 32}{4 \times 3} = 24 \] Therefore, this calculation ensures that 24 full rotations are needed.

Problem-Solving in Math

Problem-solving in math involves understanding concepts and applying them effectively. In this exercise, the steps include defining the pitch, setting up the ratio, and performing the necessary calculations. Each step simplifies the overall problem. First, identify what you need to find—in this case, the number of rotations. Then, use the given values and mathematical tools (like fraction division) to solve it. Breaking down the problem into manageable parts makes the process efficient and easier to understand.