Problem 22 - College Physics

Question

A stack of printer paper is 2 inches thick and contains 500 sheets. Estimate the thickness of an individual sheet of the printer paper. Express your answer in micrometers.

Step-by-Step Solution

Answer

The thickness of one sheet is approximately 101.6 micrometers.

Step 1: Determine Thickness in Inches

First, calculate the thickness of one sheet of paper by dividing the total thickness of the stack by the number of sheets in the stack. The total thickness of the stack is 2 inches and there are 500 sheets, so each sheet is \( \frac{2}{500} \) inches thick.

Step 2: Convert Inch Thickness to Micrometers

To convert the thickness of one sheet from inches to micrometers, note that 1 inch is equal to 25,400 micrometers. Multiply the thickness of one sheet in inches (\( \frac{1}{250} \) inches) by 25,400 micrometers/inch to find the thickness in micrometers. This calculation results in \( \frac{1}{250} \times 25,400 \) micrometers.

Step 3: Calculate Final Thickness in Micrometers

Perform the calculations set up in the previous steps: \( \frac{1}{250} \times 25,400 = 101.6 \) micrometers. Therefore, the thickness of one sheet of printer paper is approximately 101.6 micrometers.

Key Concepts

Understanding Measurement

Measurement is a fundamental concept in mathematics and science that involves determining the size, quantity, or degree of something using established units. In this example, we are looking at thickness as a type of measurement, specifically measuring how thick a stack of paper is. Measurements play a vital role in everyday problem-solving situations where precise information is needed. In mathematics, units of length such as inches and micrometers often help us describe dimensions in a way that is easy to understand.

It's essential to be familiar with different units of measurement like inches and understand how they relate to other units such as micrometers. For instance, when dealing with very small dimensions like the thickness of a sheet of paper, micrometers provide a more practical measure than inches.

Importance of Estimation

Estimation is a crucial skill in mathematics that allows us to make close approximations when exact calculations might be complex or unnecessary. In the exercise, estimating the thickness of a single sheet involves simplifying an everyday problem into manageable calculations.

We start by approximating when dividing the total thickness of 2 inches by the total number of sheets (500). This results in an estimated thickness per sheet, which can later be used in calculations involving unit conversion. It's important to note that while estimation is not exact, it is usually sufficiently close to help us make practical decisions and solve problems quickly and efficiently. Practicing estimation can also enhance number sense and help in evaluating the reasonableness of solutions.

Strategies for Effective Problem-Solving

Problem-solving in math involves using a structured approach to find solutions to various tasks. In this exercise, determining the thickness of a sheet of paper requires breaking down the task into simpler steps.

Here are common problem-solving strategies:

• Identify and understand the problem: This means knowing that we need to determine the thickness of one sheet of paper from a stack of many.
• Devise a plan: Break the problem into smaller, manageable parts (such as calculating thickness in inches and then converting units).
• Implement the solution: Carry out the calculations step-by-step as outlined in the solution.
• Review: Check if the calculated thickness makes sense by comparing with standard measurements or using logic.

Understanding and applying these strategies can significantly enhance the way we engage with mathematical problems.

Mathematics Calculations

Mathematical calculations are the backbone of solving quantitative problems. In this task, precise calculations allow us to convert a raw measurement into an understandable format.

The first calculation involves finding the thickness of one sheet of paper. Using division, we divide the total thickness by the number of sheets. After obtaining this fraction, another calculation is necessary for unit conversion. We multiply the thickness in inches by the conversion factor between inches and micrometers (25,400 micrometers per inch).

Understanding how to perform these calculations, using multiplication and division, will enhance skills in executing precise measurement conversions. It also helps in maintaining accuracy rather than relying solely on estimation. Mastery of these basic mathematical operations is critical for students to solve various problems effectively across different subjects.