Calculus help!!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
change logs to numbers.
1E6(1+-.02)
now, go to logs.
remember that log(ab)=log a+logb
Intensity=6 +log.98 to 6 +log(1.02)
The error is the difference, so
error=log1.02-log.98= .009+.009=.018
You can work it more accurately. Now differentials:
Error=highentensity-lowintensity
dError=dHigh-dlow=log1.02-log.98
To find the error in the intensity of the earthquake, we will use differentials.
The formula for the Richter scale is given by:
I = 10^a * 10^bM,
where I is the intensity of the earthquake, M is the magnitude on the Richter scale, and a and b are constants.
Let's denote the error in magnitude as ΔM and the error in intensity as ΔI. We want to find ΔI.
Differentiating both sides of the equation with respect to M:
dI = (d(10^(a + bM)) / dM,
Recall that d(10^(a + bM)) / dM = (10^(a + bM)) * ln(10) * b = b * 10^(a + bM) * ln(10).
Now, let's find ΔI:
ΔI = dI = b * 10^(a + bM) * ln(10) * ΔM.
The error in the intensity of the earthquake, ΔI, is calculated as the product of the error in magnitude, ΔM, and b * 10^(a + bM) * ln(10).
Since the problem states that the error in magnitude is at most 2%, we can say that ΔM = 0.02 * M.
Therefore, the error in the intensity of the earthquake, ΔI, is:
ΔI = b * 10^(a + bM) * ln(10) * ΔM
= b * 10^(a + bM) * ln(10) * (0.02 * M).
Please note that the values of a and b depend on the specific formula being used for the Richter scale.