Problem 58
Question
The field of view of a microscope is \((4)\left(10^{-4}\right)\) meters. If a single cell organism occupies \(\frac{1}{5}\) of the field of view, find the length of the organism in meters. Express the result in scientific notation.
Step-by-Step Solution
Verified Answer
The length of the organism is \(8 \times 10^{-5}\) meters.
1Step 1: Identify the Given Values
We are given the total field of view of a microscope as \(4 \times 10^{-4}\) meters, and the organism occupies \(\frac{1}{5}\) of this field of view.
2Step 2: Set Up the Calculation
To find the length of the organism, multiply the field of view by \(\frac{1}{5}\) since the organism occupies that fraction of the view.
3Step 3: Perform the Multiplication
Calculate \(\frac{1}{5} \times 4 \times 10^{-4}\). Start by multiplying the fraction \(\frac{1}{5}\) by 4, resulting in \(\frac{4}{5}\).
4Step 4: Express the Result in Scientific Notation
Multiply \(\frac{4}{5}\) by \(10^{-4}\) to find the length of the organism: \(\frac{4}{5} \times 10^{-4} = 0.8 \times 10^{-4}\).
5Step 5: Final Adjustment to Scientific Notation
To express \(0.8 \times 10^{-4}\) in proper scientific notation, rewrite it as \(8 \times 10^{-5}\), keeping the coefficient between 1 and 10.
Key Concepts
Field of View CalculationMicroscope MeasurementsFractional Occupation in Measurements
Field of View Calculation
The field of view of a microscope is the area visible through the microscope lenses. It's like using a camera viewfinder; however, instead of taking pictures, you see microscopic elements. Understanding the size of this field is crucial because it allows you to estimate sizes of objects you see, like tiny cells or organisms.
The field of view in this exercise is given as \(4 \times 10^{-4}\) meters. This is already expressed in scientific notation, which means it is written as a number between 1 and 10 multiplied by a power of 10. This method of writing numbers allows us to easily handle very large or very small numbers, which are common in scientific calculations.
The field of view provides essential reference points for determining the actual size of objects, by comparing how much of this known area they occupy.
The field of view in this exercise is given as \(4 \times 10^{-4}\) meters. This is already expressed in scientific notation, which means it is written as a number between 1 and 10 multiplied by a power of 10. This method of writing numbers allows us to easily handle very large or very small numbers, which are common in scientific calculations.
The field of view provides essential reference points for determining the actual size of objects, by comparing how much of this known area they occupy.
Microscope Measurements
When using a microscope, measurements are critical to understanding the sizes of the organisms or objects you're observing. The field of view acts as a ruler, against which these objects can be measured.
In our exercise, we know that the organism occupies \(\frac{1}{5}\) of the field of view. Multiplying this fraction by the total area of the field of view, \(4 \times 10^{-4}\) meters, will get us the size of the organism in meters. Calculating this way is practical because it uses proportions to translate a visible observation (how much space it takes in the view) to a specific measurement in meters.
This approach is fundamental in microscopy and allows you to convert what you see through the lenses into measurable data.
In our exercise, we know that the organism occupies \(\frac{1}{5}\) of the field of view. Multiplying this fraction by the total area of the field of view, \(4 \times 10^{-4}\) meters, will get us the size of the organism in meters. Calculating this way is practical because it uses proportions to translate a visible observation (how much space it takes in the view) to a specific measurement in meters.
This approach is fundamental in microscopy and allows you to convert what you see through the lenses into measurable data.
Fractional Occupation in Measurements
The concept of fractional occupation is straightforward: it represents how much of a total something occupies. For instance, if you have a pie and eat \(\frac{1}{5}\) of it, this fraction indicates the part of the pie you consumed.
In scientific measurements, especially with microscopes, fractional occupation helps to deduce the size of an object by knowing what fraction of the field of view it occupies. In this exercise, the organism takes up \(\frac{1}{5}\) of the field.
To find the organism's length, we calculate \(\frac{1}{5} \times 4 \times 10^{-4}\), simplifying it to \(0.8 \times 10^{-4}\). For proper scientific notation, we adjust this to \(8 \times 10^{-5}\) meters, ensuring the multiplication factor is between 1 and 10.
This final result gives a numerical representation of the organism's length, using its fractional occupation of the field of view.
In scientific measurements, especially with microscopes, fractional occupation helps to deduce the size of an object by knowing what fraction of the field of view it occupies. In this exercise, the organism takes up \(\frac{1}{5}\) of the field.
To find the organism's length, we calculate \(\frac{1}{5} \times 4 \times 10^{-4}\), simplifying it to \(0.8 \times 10^{-4}\). For proper scientific notation, we adjust this to \(8 \times 10^{-5}\) meters, ensuring the multiplication factor is between 1 and 10.
This final result gives a numerical representation of the organism's length, using its fractional occupation of the field of view.
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