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 and let R be the magnitude. We are given that R=6 and that the error in R is at most 2%, so |dR/R|≤0.02. We want to find dI, the error in I.

Using logarithmic differentiation, we have:

dR/R = ln(10) dI/I

Solving for dI, we get:

dI = (R/ln(10)) dR = (6/ln(10)) (0.02)(11eb12b0_a9ad_b619_b394_b39ba8148088_TB7866_11)

Using a calculator, we get:

dI ≈ 35

Rounding to the nearest integer, we get dI ≈ 35, so the answer is choice C.

On the Richter Scale, the Magnitude R of an Earthquake