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Study Guides > Intermediate Algebra

Read: Problem Solving With Scientific Notation

Learning Objectives

  • Solve application problems involving scientific notation
Molecule of water with one oxygen bonded to two hydrogen. Water Molecule

Solve application problems

Learning rules for exponents seems pointless without context, so let's explore some examples of using scientific notation that involve real problems. First, let's look at an example of how scientific notation can be used to describe real measurements.

Think About It

Match each length in the table with the appropriate number of meters described in scientific notation below. Write your ideas in the textboxes provided before you look at the solution.
The height of a desk Diameter of water molecule Diameter of Sun at its equator
Distance from Earth to Neptune Diameter of Earth at the Equator Height of Mt. Everest (rounded)
Diameter of average human cell Diameter of a large grain of sand Distance a bullet travels in one second
Power of 10, units in meters Length from table above
[latex]10^{12}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{9}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{6}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{4}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{2}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{0}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{-3}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{-5}[/latex]  [practice-area rows="1"][/practice-area]
[latex]10^{-10}[/latex]  [practice-area rows="1"][/practice-area]

Answer:

Power of 10, units in meters Length from table above
[latex]10^{12}[/latex] Distance from Earth to Neptune
[latex]10^{9}[/latex] Diameter of Sun at it's Equator
[latex]10^{6}[/latex] Diameter of Earth at the Equator
[latex]10^{4}[/latex] Height of Mt. Everest (rounded)
[latex]10^{2}[/latex] Distance a bullet travels in one second
[latex]10^{0}[/latex] The height of a desk
[latex]10^{-3}[/latex] Diameter of a large grain of sand
[latex]10^{-5}[/latex] Diameter of average human cell
[latex]10^{-10}[/latex] Diameter of water molecule

Red Blood Cells. Red Blood Cells
One of the most important parts of solving a "real" problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help. Here's an example that requires you to find the density of a cell, given its mass and volume. Cells aren't visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.    

Example

Human cells come in a wide variety of shapes and sizes. The mass of an average human cell is about [latex]2\times10^{-11}[/latex] grams[footnote]Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https://en.wikipedia.org/wiki/Orders_of_magnitude_(mass)[/footnote]Red blood cells are one of the smallest types of cells[footnote]How Big is a Human Cell?[/footnote], clocking in at a volume of approximately [latex]10^{-6}\text{ meters }^3[/latex].[footnote]How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May [latex]26, 2016[/latex], from http://www.weizmann.ac.il/plants/Milo/images/humanCellSize120116Clean.pdf[/footnote] Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. [footnote]Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., & Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073/pnas.1104651108[/footnote] Density is calculated as the ratio of [latex]\frac{\text{ mass }}{\text{ volume }}[/latex]. Calculate the density of an average human cell.

Answer: Read and Understand: We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell. Define and Translate:  [latex]m=\text{mass}=2\times10^{-11}[/latex], [latex]v=\text{volume}=10^{-6}\text{ meters}^3[/latex], [latex]\text{density}=\frac{\text{ mass }}{\text{ volume }}[/latex] Write and Solve: Use the quotient rule to simplify the ratio.

[latex]\begin{array}{c}\text{ density }=\frac{2\times10^{-11}\text{ grams }}{10^{-6}\text{ meters }^3}\\\text{ }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=2\times10^{-11-\left(-6\right)}\frac{\text{ grams }}{\text{ meters }^3}\\\text{ }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=2\times10^{-5}\frac{\text{ grams }}{\text{ meters }^3}\\\end{array}[/latex]

If scientists know the density of healthy cells, they can compare the density of a sick person's cells to that to rule out or test for disorders or diseases that may affect cellular density.

Answer

The average density of a human cell is [latex]2\times10^{-5}\frac{\text{ grams }}{\text{ meters }^3}[/latex]

The following video provides an example of how to find the number of operations a computer can perform in a very short amount of time. https://youtu.be/Cbm6ejEbu-o
Earth in the foreground, sun in the background, light beams traveling from the sun to the earth. Light traveling from the sun to the earth.
In the next example, you will use another well known formula, [latex]d=r\cdot{t}[/latex], to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.    

Example

The speed of light is [latex]3\times10^{8}\frac{\text{ meters }}{\text{ second }}[/latex]. If the sun is [latex]1.5\times10^{11}[/latex] meters from earth, how many seconds does it take for sunlight to reach the earth?  Write your answer in scientific notation.

Answer: Read and Understand: We are looking for how long—an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a [latex]d=r\cdot{t}[/latex] problem. Define and Translate: 

[latex]\begin{array}{l}d=1.5\times10^{11}\\r=3\times10^{8}\frac{\text{ meters }}{\text{ second }}\\t=\text{ ? }\end{array}[/latex]

Write and Solve: Substitute the values we are given into the [latex]d=r\cdot{t}[/latex] equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.

[latex]\begin{array}{c}d=r\cdot{t}\\1.5\times10^{11}=3\times10^{8}\cdot{t}\end{array}[/latex]

Divide both sides of the equation by [latex]3\times10^{8}[/latex] to isolate t.

[latex]\begin{array}{c}1.5\times10^{11}=3\times10^{8}\cdot{t}\\\text{ }\\\frac{1.5\times10^{11}}{3\times10^{8}}=\frac{3\times10^{8}}{3\times10^{8}}\cdot{t}\end{array}[/latex]

On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with t. 

[latex]\begin{array}{c}\frac{1.5\times10^{11}}{3\times10^{8}}=\frac{3\times10^{8}}{3\times10^{8}}\cdot{t}\\\text{ }\\\left(\frac{1.5}{3}\right)\times\left(\frac{10^{11}}{10^{8}}\right)=t\\\text{ }\\\left(0.5\right)\times\left(10^{11-8}\right)=t\\0.5\times10^3=t\end{array}[/latex]

This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of [latex]10[/latex].

[latex]0.5\times10^3=5.0\times10^2=t[/latex]

Answer

The time it takes light to travel from the sun to the earth is [latex]5.0\times10^2=t[/latex] seconds, or in standard notation, [latex]500[/latex] seconds.  That's not bad considering how far it has to travel!

In the following video you will see that the total number of miles run by the participants in the Boston marathon is larger than the circumference of the earth! https://youtu.be/san2avgwu6k

Summary

Scientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of [latex]10[/latex]. The format is written [latex]a\times10^{n}[/latex], where [latex]1\leq{a}<10[/latex] and n is an integer. To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.

Licenses & Attributions

CC licensed content, Original

  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
  • Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Application of Scientific Notation - Quotient 2 (Time for Computer Operations). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.