Question
• About how much longer is the diameter of a grain of salt than the width of a human hair Grain of Salt (diameter) 3x10^-4 Human Hair (width) 8x10^-5
Answers
GPT-4o mini
To compare the diameter of a grain of salt to the width of a human hair, we need to divide the diameter of the grain of salt by the width of the human hair.
Given:
- Diameter of a grain of salt = \(3 \times 10^{-4}\) meters
- Width of a human hair = \(8 \times 10^{-5}\) meters
Now, we can calculate the ratio:
\[
\text{Ratio} = \frac{\text{Diameter of grain of salt}}{\text{Width of human hair}} = \frac{3 \times 10^{-4}}{8 \times 10^{-5}}
\]
To simplify this, we can calculate it as follows:
\[
\frac{3}{8} \times \frac{10^{-4}}{10^{-5}} = \frac{3}{8} \times 10^{1} = \frac{3}{8} \times 10 = \frac{30}{8} = 3.75
\]
Thus, the diameter of a grain of salt is about **3.75 times longer** than the width of a human hair.
Given:
- Diameter of a grain of salt = \(3 \times 10^{-4}\) meters
- Width of a human hair = \(8 \times 10^{-5}\) meters
Now, we can calculate the ratio:
\[
\text{Ratio} = \frac{\text{Diameter of grain of salt}}{\text{Width of human hair}} = \frac{3 \times 10^{-4}}{8 \times 10^{-5}}
\]
To simplify this, we can calculate it as follows:
\[
\frac{3}{8} \times \frac{10^{-4}}{10^{-5}} = \frac{3}{8} \times 10^{1} = \frac{3}{8} \times 10 = \frac{30}{8} = 3.75
\]
Thus, the diameter of a grain of salt is about **3.75 times longer** than the width of a human hair.
Explain
GPT-4o mini
Certainly! Let's break down the comparison between the diameter of a grain of salt and the width of a human hair step-by-step.
1. **Understanding the Measurements**:
- The diameter of a grain of salt is given as \(3 \times 10^{-4}\) meters. This means it is 0.0003 meters, or 0.3 millimeters.
- The width of a human hair is given as \(8 \times 10^{-5}\) meters. This translates to 0.00008 meters, or 0.08 millimeters.
2. **Setting Up the Ratio**:
- To find out how much longer the grain of salt's diameter is than the human hair's width, you can set up a ratio:
\[
\text{Ratio} = \frac{\text{Diameter of grain of salt}}{\text{Width of human hair}} = \frac{3 \times 10^{-4}}{8 \times 10^{-5}}
\]
3. **Dividing the Numbers**:
- When you divide the two measurements, you can handle the coefficients (the numbers in front) and the powers of ten separately.
- First, divide the coefficients: \(3\) (from the grain of salt) divided by \(8\) (from the human hair) gives:
\[
\frac{3}{8} = 0.375
\]
- Next, divide the powers of ten:
\(\frac{10^{-4}}{10^{-5}} = 10^{-4 - (-5)} = 10^{-4 + 5} = 10^{1}\), which is \(10\).
4. **Combining the Results**:
- Now you multiply the results together:
\[
0.375 \times 10 = 3.75
\]
5. **Conclusion**:
- This means that the diameter of a grain of salt is about 3.75 times larger than the width of a human hair.
So, in summary, by setting up the division of the two measurements, we calculated a ratio that shows that a grain of salt’s diameter is 3.75 times longer than a human hair’s width.
1. **Understanding the Measurements**:
- The diameter of a grain of salt is given as \(3 \times 10^{-4}\) meters. This means it is 0.0003 meters, or 0.3 millimeters.
- The width of a human hair is given as \(8 \times 10^{-5}\) meters. This translates to 0.00008 meters, or 0.08 millimeters.
2. **Setting Up the Ratio**:
- To find out how much longer the grain of salt's diameter is than the human hair's width, you can set up a ratio:
\[
\text{Ratio} = \frac{\text{Diameter of grain of salt}}{\text{Width of human hair}} = \frac{3 \times 10^{-4}}{8 \times 10^{-5}}
\]
3. **Dividing the Numbers**:
- When you divide the two measurements, you can handle the coefficients (the numbers in front) and the powers of ten separately.
- First, divide the coefficients: \(3\) (from the grain of salt) divided by \(8\) (from the human hair) gives:
\[
\frac{3}{8} = 0.375
\]
- Next, divide the powers of ten:
\(\frac{10^{-4}}{10^{-5}} = 10^{-4 - (-5)} = 10^{-4 + 5} = 10^{1}\), which is \(10\).
4. **Combining the Results**:
- Now you multiply the results together:
\[
0.375 \times 10 = 3.75
\]
5. **Conclusion**:
- This means that the diameter of a grain of salt is about 3.75 times larger than the width of a human hair.
So, in summary, by setting up the division of the two measurements, we calculated a ratio that shows that a grain of salt’s diameter is 3.75 times longer than a human hair’s width.