Question

You are given these approximate measurements: (a) the radius of Earth is $6 \times 10^{6} \mathrm{m},\( (b) the length of a human body is \)6 \mathrm{ft},(\mathrm{c})\( a cell's diameter is \)10^{-6} \mathrm{m},$ (d) the width of the hemoglobin molecule is \(3 \times 10^{-9} \mathrm{m},\) and (e) the distance between two atoms (carbon and nitrogen) is $3 \times 10^{-10} \mathrm{m} .$ Write these measurements in the simplest possible metric prefix forms (in either nm, Mm, \(\mu \mathrm{m}\), or whatever works best).

Short answer

Question: Rewrite the following measurements using the most appropriate metric prefixes: (a) The radius of Earth (b) The length of a human body (c) A cell's diameter (d) The width of the hemoglobin molecule (e) The distance between two atoms (carbon and nitrogen) Answer: (a) 6 Mm (b) 1.83 m (c) 1 µm (d) 3 nm (e) 0.3 nm

Step-by-step solution

Review metric prefixes and their powers of ten

We will use the following metric prefixes: - nano (n): \(10^{-9}\) - micro (µ): \(10^{-6}\) - milli (m): \(10^{-3}\) - base unit (no prefix): \(10^0\) - kilo (k): \(10^3\) - Mega (M): \(10^6\) - Giga (G): \(10^9\)

Rewrite the radius of Earth in the most appropriate metric prefix

Given: The radius of Earth is \(6 \times 10^6 \, \mathrm{m}\). Since this is in the base unit (meters), we can use Mega (M) to simplify the measurement: \(6 \times 10^6 \, \mathrm{m} = 6 \, \mathrm{Mm}\) (Mega meters)

Rewrite the length of a human body in the most appropriate metric prefix

Given: The length of a human body is \(6 \, \mathrm{ft}\). First, we need to convert feet to meters. There are approximately 0.3048 meters in 1 foot. So, \(6 \, \mathrm{ft} \times 0.3048 \, \frac{\mathrm{m}}{\mathrm{ft}} \approx 1.83\, \mathrm{m}\). Since this value is close to the base unit, there is no need to use a metric prefix.

Rewrite a cell's diameter in the most appropriate metric prefix

Given: A cell's diameter is \(10^{-6} \, \mathrm{m}\). Since this measurement is in meters, we can use micro (µ) to simplify the measurement: \(10^{-6} \, \mathrm{m} = 1 \, \mu \mathrm{m}\) (micro meters)

Rewrite the width of the hemoglobin molecule in the most appropriate metric prefix

Given: The width of the hemoglobin molecule is \(3 \times 10^{-9} \, \mathrm{m}\). Since this measurement is in meters, we can use nano (n) to simplify the measurement: \(3 \times 10^{-9} \, \mathrm{m} = 3 \, \mathrm{nm}\) (nano meters)

Rewrite the distance between two atoms (carbon and nitrogen) in the most appropriate metric prefix

Given: The distance between two atoms (carbon and nitrogen) is \(3 \times 10^{-10} \, \mathrm{m}\). This value is between the base and nano meters, so, we can rewrite it as a multiple of nanometers: \(3 \times 10^{-10} \, \mathrm{m} = 0.3 \, \mathrm{nm}\) (nano meters) In conclusion, we have rewritten the given measurements using the most appropriate metric prefixes: (a) The radius of Earth: 6 Mm (b) The length of a human body: 1.83 m (c) A cell's diameter: 1 µm (d) The width of the hemoglobin molecule: 3 nm (e) The distance between two atoms (carbon and nitrogen): 0.3 nm