Answer :
### Solution:
Let's analyze the problem step-by-step:
#### (a) Calculate the earthquake reading on the seismograph
We are given that:
- Magnitude of the first earthquake ([tex]\(M_1\)[/tex]) = 7.7
- Magnitude of the second earthquake ([tex]\(M_2\)[/tex]) = 5.7
The formula to measure the magnitude of an earthquake based on seismograph readings is:
[tex]\[ M(x) = \log_{10}\left(\frac{x}{0.001}\right) \][/tex]
To find the seismograph reading ([tex]\(x\)[/tex]) for a given magnitude ([tex]\(M\)[/tex]), we rearrange the formula:
[tex]\[ M(x) = \log_{10}\left(\frac{x}{0.001}\right) \][/tex]
[tex]\[ 10^{M(x)} = \frac{x}{0.001} \][/tex]
[tex]\[ x = 0.001 \times 10^{M(x)} \][/tex]
Let's calculate [tex]\( x \)[/tex] for both magnitudes:
1. For [tex]\( M_1 = 7.7 \)[/tex]:
[tex]\[ x_1 = 0.001 \times 10^{7.7} \][/tex]
[tex]\[ x_1 \approx 50118.723 \text{ millimeters} \][/tex]
2. For [tex]\( M_2 = 5.7 \)[/tex]:
[tex]\[ x_2 = 0.001 \times 10^{5.7} \][/tex]
[tex]\[ x_2 \approx 501.187 \text{ millimeters} \][/tex]
#### Compare the readings
Now, let's compare the seismograph readings:
[tex]\[ \text{Comparison} = \frac{x_1}{x_2} \][/tex]
Substituting the values:
[tex]\[ \text{Comparison} = \frac{50118.723}{501.187} \][/tex]
[tex]\[ \approx 100 \][/tex]
This result shows that the first earthquake (magnitude 7.7) produces a seismograph reading approximately 100 times greater than the second earthquake (magnitude 5.7).
#### Conclusion on the magnitudes of earthquakes
From the analysis, we can conclude that:
- A magnitude 7.7 earthquake is significantly more powerful than a magnitude 5.7 earthquake, with the seismograph reading indicating it is about 100 times stronger.
- This significant difference illustrates the logarithmic nature of the Richter scale, with each whole number increase representing a tenfold increase in amplitude of the seismic waves, and roughly 31.6 times more energy release.
#### (b) Why experts use the logarithmic function
Experts use the logarithmic function for several reasons:
1. Range Representation: Earthquake magnitudes vary over a wide range of values. A logarithmic scale compresses this range into manageable numbers, making it easier to compare magnitudes.
2. Human Perception: The logarithmic scale is more intuitive in reflecting the human perception of the intensity of natural phenomena, which often also follows a logarithmic relationship.
3. Energy Release: The energy released by earthquakes grows exponentially with magnitude. A logarithmic scale directly aligns with the exponential increase in energy, providing a more meaningful representation of the earthquake's impact.
Therefore, the logarithmic function is extremely useful in representing and comparing the enormous differences in energy release and amplitudes of seismic waves associated with earthquakes.
By analyzing the data and comprehensively interpreting the results, we get a deeper understanding of the severity of earthquakes on the Richter scale and why logarithmic functions are suitable for such measurements.
Let's analyze the problem step-by-step:
#### (a) Calculate the earthquake reading on the seismograph
We are given that:
- Magnitude of the first earthquake ([tex]\(M_1\)[/tex]) = 7.7
- Magnitude of the second earthquake ([tex]\(M_2\)[/tex]) = 5.7
The formula to measure the magnitude of an earthquake based on seismograph readings is:
[tex]\[ M(x) = \log_{10}\left(\frac{x}{0.001}\right) \][/tex]
To find the seismograph reading ([tex]\(x\)[/tex]) for a given magnitude ([tex]\(M\)[/tex]), we rearrange the formula:
[tex]\[ M(x) = \log_{10}\left(\frac{x}{0.001}\right) \][/tex]
[tex]\[ 10^{M(x)} = \frac{x}{0.001} \][/tex]
[tex]\[ x = 0.001 \times 10^{M(x)} \][/tex]
Let's calculate [tex]\( x \)[/tex] for both magnitudes:
1. For [tex]\( M_1 = 7.7 \)[/tex]:
[tex]\[ x_1 = 0.001 \times 10^{7.7} \][/tex]
[tex]\[ x_1 \approx 50118.723 \text{ millimeters} \][/tex]
2. For [tex]\( M_2 = 5.7 \)[/tex]:
[tex]\[ x_2 = 0.001 \times 10^{5.7} \][/tex]
[tex]\[ x_2 \approx 501.187 \text{ millimeters} \][/tex]
#### Compare the readings
Now, let's compare the seismograph readings:
[tex]\[ \text{Comparison} = \frac{x_1}{x_2} \][/tex]
Substituting the values:
[tex]\[ \text{Comparison} = \frac{50118.723}{501.187} \][/tex]
[tex]\[ \approx 100 \][/tex]
This result shows that the first earthquake (magnitude 7.7) produces a seismograph reading approximately 100 times greater than the second earthquake (magnitude 5.7).
#### Conclusion on the magnitudes of earthquakes
From the analysis, we can conclude that:
- A magnitude 7.7 earthquake is significantly more powerful than a magnitude 5.7 earthquake, with the seismograph reading indicating it is about 100 times stronger.
- This significant difference illustrates the logarithmic nature of the Richter scale, with each whole number increase representing a tenfold increase in amplitude of the seismic waves, and roughly 31.6 times more energy release.
#### (b) Why experts use the logarithmic function
Experts use the logarithmic function for several reasons:
1. Range Representation: Earthquake magnitudes vary over a wide range of values. A logarithmic scale compresses this range into manageable numbers, making it easier to compare magnitudes.
2. Human Perception: The logarithmic scale is more intuitive in reflecting the human perception of the intensity of natural phenomena, which often also follows a logarithmic relationship.
3. Energy Release: The energy released by earthquakes grows exponentially with magnitude. A logarithmic scale directly aligns with the exponential increase in energy, providing a more meaningful representation of the earthquake's impact.
Therefore, the logarithmic function is extremely useful in representing and comparing the enormous differences in energy release and amplitudes of seismic waves associated with earthquakes.
By analyzing the data and comprehensively interpreting the results, we get a deeper understanding of the severity of earthquakes on the Richter scale and why logarithmic functions are suitable for such measurements.
