Chapter 1 | Functions and Graphs
107
⎛ ⎝ A 1 A 2
⎞ ⎠ .
8−7= log 10
Therefore,
⎛ ⎝ A 1 A 2
⎞ ⎠ =1,
log 10
which implies A 1 / A 2 =10 or A 1 =10 A 2 . Since A 1 is 10 times the size of A 2 , we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation log 10 ⎛ ⎝ A 1 A 2 ⎞ ⎠ =8−6=2. Therefore, A 1 =100 A 2 . That is, the first earthquake is 100 times more intense than the second earthquake. How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?
Solution To compare the Japan and Haiti earthquakes, we can use an equation presented earlier: 9−7.3= log 10 ⎛ ⎝ A 1 A 2 ⎞ ⎠ . Therefore, A 1 / A 2 =10
1.7 , and we conclude that the earthquake in Japan was approximately 50 times more
intense than the earthquake in Haiti.
Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.
1.33
Hyperbolic Functions The hyperbolic functions are defined in terms of certain combinations of e x and e − x . These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ( Figure 1.49 ). If we introduce a coordinate system so that the low point of the chain lies along the y -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions .
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