Richter Scale and Exponential Functions

TENRM Math 102 - Spring 2001 – May 31

 

Solutions are on the back of this sheet. You do not have to hand in the answers to these questions this quarter. We will revisit this topic in the Fall.

 

The Richter Scale follows a logarithmic scale.  See your Understand Earth text for more explanation of how the Richter scale is used in Geology. The Richter Scale is a logarithmic function that uses the common base, 10. The equation is:

 

                              M = 10 R                  or                  R = log M

Where M is the amount of ground movement (strength of the earthquake) And R is the strength of the earthquake on the Richter Scale. Because the Richter Scale uses a base of 10 each increase of 1 in the strength of an earthquake on the Richter Scale means that there is 10 times as much ground movement.

 

The amount of energy in an earthquake increases by a factor of 1000 (also called 3 magnitudes) when the value on the Richter Scale increases by 2.  The equation for the energy in an earthquake is

                                                                           E » 31R

 

Where E is the factor the energy increases by and R is the increase in the value on the Richter scale. For example, there is 1000 times more energy in an earthquake that measures 6.8 than one that measures 4.8 on the Richter Scale.

 

Use this information to answer the following questions

1)     How much more ground movement is there in an earthquake that registers 5 on the Richter Scale than one that registers 3?

 

2)     How much more energy is there in an earthquake that registers 5 on the Richter Scale than one that registers 3?

 

3)     How much more ground movement is there in an earthquake that registers 7 on the Richter Scale than one that registers 3?

 

4)     How much more energy is there in an earthquake that registers 7 on the Richter Scale than one that registers 3?

 

5)     The 1906 earthquake in San Francisco registered 8.3 on the Richter Scale. How much more ground movement and how much more energy was released in that earthquake than a 4.3 earthquake?

 

6)     The strongest earthquake listed in table 18.1 is 8.9 on the Richter Scale. How much more ground movement and how much more energy is this than the 1906 San Francisco earthquake?

 

7)     Earthquake A moves the ground 50,000 times more than earthquake B. What is the difference on the Richter Scale?

 

8)     Earthquake C is 25,0000,000 times weaker (less ground movement) than earthquake D. What is their difference on the Richter Scale?

 

9)     The weakest earthquake commonly felt is a 3 on the Richter Scale. How much more energy was released in the 1906 San Francisco earthquake than an earthquake registering 3 on the Richter Scale?

 

10)   What would be the Richter scale reading of an earthquake than moves the ground 1 million times more than the San Francisco earthquake?


Solutions        Richter Scale

TENRM Math 102 - Spring 2001 – may 31

Solutions are in italics

1)     How much more ground movement is there in an earthquake that registers 5 on the Richter Scale than one that registers 3?

Solution:An increase in 2 in the Richter Scale means an increase of 102 or 100  in the ground movement

 

2)     How much more energy is there in an earthquake that registers 5 on the Richter Scale than one that registers 3?

Solution: An increase in 2 in the Richter Scale means an increase of 312 or 1000  in the energy.

 

3)     How much more ground movement is there in an earthquake that registers 7 on the Richter Scale than one that registers 3?

Solution: An increase in 4 in the Richter Scale means an increase of 104 or 10000  in the ground movement

 

4)     How much more energy is there in an earthquake that registers 7 on the Richter Scale than one that registers 3?

Solution: An increase in 4 in the Richter Scale means an increase of 314 or 1 million  in the energy.

 

5)     The 1906 earthquake in San Francisco registered 8.3 on the Richter Scale. How much more ground movement and how much more energy was released in that earthquake than a 4.3 earthquake?

Solution: An increase in 4 in the Richter Scale means an increase of 104 or 10000  in the ground movement and 314 or 1 million  in the energy.

 

6)     The strongest earthquake listed in table 18.1 is 8.9 on the Richter Scale. How much more ground movement and how much more energy is this than the 1906 San Francisco earthquake?

Solution: An increase from 8.3 to 8.9, or 0.6, means an increase of 100.6 = 3.98 times more ground movement and 310.6 = 7.85 times more energy

 

7)     Earthquake A moves the ground 50,000 times more than earthquake B. What is the difference on the Richter Scale? 

Solution: Use the relationship R = log M.   R = log (50,000) = Earthquake A registers 4.70 more on the Richter Scale than earthquake B.

 

8)     Earthquake C is 25,0000,000 times weaker (less ground movement) than earthquake D. What is their difference on the Richter Scale?

Solution:  R = log (250000000) =  Earthquake C registers 8.40 less on the Richter Scale than earthquake D.

 

9)     The weakest earthquake commonly felt is a 3 on the Richter Scale. How much more energy was released in the 1906 San Francisco earthquake than an earthquake registering 3 on the Richter Scale?

Solution: Since the difference between the two earthquakes is 8.3 - 3 = 5.3 then the difference in energy is E = 105.3 = 199,500 times more energy in the 1906 earthquake.

 

10)   What would be the Richter scale reading of an earthquake than moves the ground 1 million times more than the San Francisco earthquake?

Solution: 1 million times more ground movement means a Richter Scale reading of R = log(1,0000,000) = 6 more, or  8.3 + 6 = 14.3 on the Richter Scale. (Note that no earthquake has e