On the Richter scale, the magnitude R of an earthquake is given by the formula where I is the intensity of the earthquake being measured and is the standard reference intensity. Suppose an earthquake is measured with a magnitude of 6 on the Richter scale with an error of at most 2%. Use differentials to find the error in the intensity of the earthquake. Round your answer to the nearest integer.

Question

On the Richter scale, the magnitude R of an earthquake is given by the formula  where I is the intensity of the earthquake being measured and  is the standard reference intensity. Suppose an earthquake is measured with a magnitude of 6 on the Richter scale with an error of at most 2%. Use differentials to find the error in the intensity of the earthquake. Round your answer to the nearest integer.

Quizplus · Accepted Answer

Let I be the intensity of the earthquake being measured. We know that R=6 and dR/R=0.02. We want to find dI, the error in the intensity of the earthquake. We have:
dR/R = ln(I/I0)ln(10)
So, taking differentials of both sides:
dR/R = (1/ln(10))(1/I)dI
Solving for dI, we get:
dI = (ln(I/I0))(ln(10))(dR/R)I
Substituting the given values, we get:
dI = (ln(10))(0.02)(I)e^(-6.0 ln(10))
dI ≈ 0.214I
Rounding this to the nearest integer gives an error of 21%. Therefore, the correct option is C, i.e., 11eaa8b1_daee_c44a_b1b6_dd80116c4bc7_TB6026_11.

On the Richter Scale, the Magnitude R of an Earthquake