Problem 45 - Basic College Mathematics

Question

The pitch of a screw is the distance between its threads. With each complete rotation, the screw goes in or out a distance equal to its pitch. Use this information. The pitch of a screw is \(\frac{1}{16}\) in. How far will it go into a piece of oak when it is turned 10 complete rotations clockwise?

Step-by-Step Solution

Answer

\(\frac{5}{8}\) inches

Step 1: Understand the Pitch

The pitch of the screw represents the distance the screw travels into the material with each complete rotation. Given that the pitch is \(\frac{1}{16}\) inch, this means with one complete rotation the screw moves \(\frac{1}{16}\) inch into the material.

Step 2: Identify the Number of Rotations

The problem states that the screw is turned 10 complete rotations.

Step 3: Calculate Total Distance

To find the total distance the screw travels into the oak piece, multiply the pitch (\(\frac{1}{16}\) inch) by the number of rotations (10).\[\frac{1}{16} \times 10 = \frac{10}{16} = \frac{5}{8} \, \text{inches}\]

Key Concepts

screw pitch

The pitch of a screw is a fundamental concept in understanding how screws work. It represents the distance between the threads of a screw. When a screw makes one complete rotation, it moves forward or backward by a distance equal to its pitch. This is crucial for precise applications where the exact movement per rotation is needed. For example, if a screw has a pitch of \(\frac{1}{16}\) inch, every turn will advance the screw 1/16 of an inch into the material.

distance calculation

To determine how far a screw will travel into a material after a certain number of rotations, we need to understand basic distance calculation. This involves multiplying the pitch by the number of rotations. For example, with a pitch of \(\frac{1}{16}\) inch and 10 rotations, the total distance traveled will be:

\[ \frac{1}{16} \times 10 = \frac{10}{16} = \frac{5}{8} \text{inches} \]

This calculation is simple but essential, as it tells you exactly how deep the screw will embed into the material after a given number of turns.

fraction multiplication

Fraction multiplication is a key mathematical skill used in distance calculations for screws. When we multiply fractions, we multiply the numerators together and the denominators together. For the example given, we multiply \(\frac{1}{16}\) by 10, treating 10 as \(\frac{10}{1}\):

\[ \frac{1}{16} \times \frac{10}{1} = \frac{1 \times 10}{16 \times 1} = \frac{10}{16} = \frac{5}{8} \] Understanding how to handle fractions is important not just for this problem, but for most kinds of mathematic and scientific calculations.

problem-solving steps

Solving any mathematical problem involves following a series of logical steps. Here's a simple breakdown you can follow:

• Read and understand the problem statement.
• Identify the given values. In this case, the pitch of the screw and the number of rotations.
• Set up the problem using a relevant mathematical operation. Here, it is multiplication.
• Perform the calculation carefully.
• Double-check your work to ensure it’s correct.

When faced with a problem, always take it step by step to avoid mistakes and understand the concept fully.