Objectives
- Reinforce your ability to convert between standard and scientific notation and interpret the meaning of powers of ten in real-world and scientific contexts.
Worksheet 1.2.1 Explore: Scientific Notation
This worksheet is designed to help you explore the concept of scientific notation in physics.
1.
The influenza virus can have a diameter of about 0.000000099 meters. Convert 0.000000099 to scientific notation.
Hint.
The decimal point would need to move to the right by 8 places.
Answer.
\(9.9 \times 10^{-8}\)
Solution.
To write 0.000000099 in scientific notation, move the decimal point eight places to the right to place it after the first nonzero digit. This gives "9.9". Since the decimal moved right by eight places, the exponent on 10 is -8.
So, \(0.000000099 = 9.9 \times 10^{-8}\text{.}\)
2.
A strand of human hair is about 0.00007 meters in diameter. Convert 0.00007 to scientific notation.
Hint.
The decimal point would need to move to the right by 5 places.
Answer.
\(7.0 \times 10^{-5}\)
Solution.
To write 0.00007 in scientific notation, move the decimal point five places to the right to place it after the first nonzero digit. This gives "7.0". Since the decimal moved right by five places, the exponent on 10 is -5.
So, \(0.00007 = 7.0 \times 10^{-5}\text{.}\)
3.
A grain of sand can be about 0.0012 meters in diameter. Convert 0.0012 to scientific notation.
Hint.
The decimal point would need to move to the right by 3 places.
Answer.
\(1.2 \times 10^{-3}\)
Solution.
To write 0.0012 in scientific notation, move the decimal point three places to the right to place it after the first nonzero digit. This gives "1.2". Since the decimal moved right by three places, the exponent on 10 is -3.
So, \(0.0012 = 1.2 \times 10^{-3}\text{.}\)
4.
An average adult human might be about 1.75 meters tall. Convert 1.75 to scientific notation.
Hint.
The decimal point would need to move to the left by 0 places.
Answer.
\(1.75 \times 10^{0}\)
Solution.
To write 1.75 in scientific notation, the decimal is already after the first nonzero digit, so we don’t need to move it. That means the exponent on 10 is 0.
So, \(1.75 = 1.75 \times 10^{0}\text{.}\)
5.
The Empire State Building in New York City is about \(3.81 \times 10^{2}\) meters tall. Convert \(3.81 \times 10^{2}\) to standard notation.
Hint.
A positive exponent means move the decimal to the right. Move it 2 places to the right.
Answer.
381
Solution.
\(3.81 \times 10^{2}\) means move the decimal point 2 places to the right: \(3.81 → 38.1 → 381.\text{.}\)
So, \(3.81 \times 10^{2} = 381\text{.}\)
6.
Mount Everest is approximately \(8.85 \times 10^{3}\) meters tall. Convert \(8.85 \times 10^{3}\) to standard notation.
Hint.
Move the decimal point 3 places to the right.
Answer.
8850
Solution.
\(8.85 \times 10^{3}\) means move the decimal 3 places to the right: \(8.85 → 88.5 → 885 → 8850\text{.}\)
So, \(8.85 \times 10^{3} = 8850\text{.}\)
7.
The Earth’s diameter is about \(1.27 \times 10^{7}\) meters. Convert \(1.27 \times 10^{7}\) to standard notation.
Hint.
Move the decimal 7 places to the right.
Answer.
12,700,000
Solution.
\(1.27 \times 10^{7}\) means move the decimal 7 places to the right: \(1.27 → 12.7 → 127 → … → 12,700,000\text{.}\)
So, \(1.27 \times 10^{7} = 12,700,000\text{.}\)
8.
The average distance from Earth to the Sun is about \(1.496 \times 10^{11}\) meters. Convert \(1.496 \times 10^{11}\) to standard notation.
Hint.
Move the decimal 11 places to the right.
Answer.
149,600,000,000
Solution.
\(1.496 \times 10^{11}\) means move the decimal 11 places to the right: \(1.496 → 14.96 → 149.6 → … → 149,600,000,000\text{.}\)
So, \(1.496 \times 10^{11} = 149,600,000,000\text{.}\)
9.
About how many orders of magnitude larger is a human compared to a grain of sand? Use questions Worksheet Exercise 1.2.1.3 and Worksheet Exercise 1.2.1.4 to help.
Hint.
Compare the exponents: \(10^{0}\) vs. \(10^{-3}\text{.}\) How many powers of ten apart are they?
Answer.
3
Solution.
The exponent for the human’s height is 0, and for the grain of sand it is -3. The difference is \(0 - (-3) = 3\text{.}\)
So, a human is approximately 3 orders of magnitude larger than a grain of sand. This means the human is about \(10^3 = 1,000\) times taller than the grain of sand.
In other words, if you stacked 1,000 grains of sand end to end, they would reach about the height of one person!
10.
About how many orders of magnitude larger is a human compared to an influenza virus? Use questions Worksheet Exercise 1.2.1.1 and Worksheet Exercise 1.2.1.4 to help.
Hint.
Compare the exponents: \(10^{0}\) vs. \(10^{-8}\text{.}\) How many powers of ten apart are they?
Answer.
8
Solution.
The exponent for the human’s height is 0, and for the virus it is -8. The difference is \(0 - (-8) = 8\text{.}\)
So, a human is approximately 8 orders of magnitude larger than the influenza virus. This means the human is about \(10^8 = 100,000,000\) times taller than the virus.
In other words, if you stacked 100,000,000 viruses end to end, they would reach about the height of one person!
