Answer :
Final answer:
To find dE/dM for different moment magnitudes M, we differentiate the given equation log₁₀(E) = 16.1 + 1.5M with respect to M, apply the chain rule for logarithmic differentiation, and substitute the values M = 2, 5, and 8 into the derived formula to calculate the rate of change of energy with respect to magnitude.
Explanation:
The relationship between the energy (E) released by an earthquake and its moment magnitude (M) is given by the equation log₁₀(E) = 16.1 + 1.5M. To find the rate at which energy changes with respect to moment magnitude, dE/dM, we take the derivative of the energy function with respect to M.
To solve for dE/dM, we apply the chain rule where E is a function of log₁₀(E). Using implicit differentiation, we differentiate both sides of the equation with respect to M:
Take the derivative of the left side using the chain rule: d/dM [log₁₀(E)] = d(log₁₀(E))/dE * dE/dM
Since the derivative of log₁₀(E) with respect to E is 1/(Eln(10)), this gives us 1/(Eln(10)) * dE/dM.
The derivative of the right side of the equation with respect to M is simple: d/dM [16.1 + 1.5M] = 1.5.
Now, equate the two expressions to solve for dE/dM: 1/(Eln(10)) * dE/dM = 1.5
Multiply both sides by Eln(10) to isolate dE/dM: dE/dM = 1.5Eln(10)
Remember that E itself is 10^{16.1+1.5M}, so substituting this into dE/dM gives us the formula for the rate of change of energy with respect to magnitude.
To calculate the specific values of dE/dM for M = 2, 5, and 8, we use these moment magnitudes in the equation:
For M=2: dE/dM = 1.5(10^{16.1 + 1.5(2)})ln(10)
For M=5: dE/dM = 1.5(10^{16.1 + 1.5(5)})ln(10)
For M=8: dE/dM = 1.5(10^{16.1 + 1.5(8)})ln(10)
Once we calculate these, we can find how quickly energy changes in response to changes in moment magnitude.
