Question

A human hair has an average diameter of \(2.5 \times 10^{-5}\) meters. What is the maximum number of human hairs that could fit side by side (without overlapping) on a microscope slide that is 50 millimeters wide? Write your answer in scientific notation. (A) \(2.0 \times 10^3\) (B) \(2.0 \times 10^5\) (C) \(2 \times 10^6\) (D) \(10^{-3}\)

Short answer

The maximum number of human hairs that could fit side by side on a microscope slide is \(2.0 \times 10^3\).

Step-by-step solution

Convert width of the microscope slide to meters

We are given that the microscope slide is 50 millimeters wide. Since there are 1000 millimeters (mm) in a meter (m), we can convert 50 mm to meters using the following ratio: \[ \text{width in meters} = \frac{50\, \text{mm}}{1000\, \text{mm/m}} = 0.05\, \text{m} \]

Divide the width of the microscope slide by the diameter of a human hair

Now that we have the width of the microscope slide in meters, we can divide this by the average diameter of a human hair, which is \(2.5 \times 10^{-5}\) meters: \[ \frac{0.05\, \text{m}}{2.5 \times 10^{-5}\, \text{m}} = \frac{0.05}{2.5 \times 10^{-5}} \]

Simplify the expression and write the answer in scientific notation

To simplify, we can multiply both the numerator and denominator by \(10^5\) which results in: \[ \frac{0.05 \times 10^5}{2.5 \times 10^{-5} \times 10^5} = \frac{5 \times 10^3}{2.5 \times 10^0} \] Now, divide 5 by 2.5, which equals 2, and subtract the exponents of 10: \[ 2 \times 10^{3-0} = 2 \times 10^3 \] The maximum number of human hairs that could fit side by side on a microscope slide is \(2.0 \times 10^3\). The correct answer is (A).

Key concepts

Unit Conversion

Converting units is a fundamental skill in science and mathematics, especially when dealing with measurements of different magnitudes. This problem begins by requiring a conversion from millimeters to meters, as the dimensions given for the microscope slide are in millimeters while the diameter of the human hair is in meters. To make calculations easier, both measurements should be in the same unit.

In this case, the conversion factor is important to remember: 1 meter equals 1000 millimeters. To convert millimeters to meters, you divide by 1000. For example, 50 millimeters becomes \[\text{meters} = \frac{50 \text{ mm}}{1000 \text{ mm/m}} = 0.05 \text{ m}.\]This simple division helps to standardize the measurements, allowing for easy comparison and further calculations. Remember: when converting, ensure that both numerator and denominator have the correct corresponding units to cancel out properly.

Dimensional Analysis

Dimensional analysis is a powerful tool in converting, simplifying, and solving problems by using the units of measurement involved. It allows you to check that your equations make sense and are consistent through unit cancellation. In this exercise, it ensures that calculations consistently use meters as units.

The process involves:

• Identifying the units involved in each measurement.
• Converting all measurements to some common unit, often the SI units like meters.
• Applying the conversion and checking that appropriate units are included or cancelled out during calculations.

Once the width of the microscope slide is converted to meters (0.05 meters), we can perform division using dimensional analysis efficiently:

\[\frac{0.05 \text{ m}}{2.5 \times 10^{-5} \text{ m}}.\]The matching units (meters) cancel each other out, leaving a unit-less answer which is vital for clarity and correctness in further calculations.

Problem Solving

Problem solving is a strategic approach to understanding and finding solutions. It involves analyzing the problem, planning the next steps, executing the plan, and confirming that the solution meets the expected outcome. This particular problem is an example of applying both unit conversion and dimensional analysis in a coherent problem-solving strategy.

The steps used here include breaking down the larger problem — determining how many human hairs fit on the slide — into smaller, manageable parts:

• Convert units to ensure consistency.
• Use dimensional analysis to maintain correctness and clarity of units.
• Simplify the results and write the final answer using scientific notation.

For example, the problem simplifies to a straightforward division after conversions:\[\frac{0.05}{2.5 \times 10^{-5}} = 2 \times 10^3.\]The result is presented in scientific notation, a concise method of expressing large or small numbers clearly, ensuring accuracy and ease of communication. Through these systematic steps, you can develop a habit of logical thinking and precision in your mathematical and scientific problem solving.