Question

The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

Accepted Answer

First, identify the function given in the problem: $$ E = 25,000 \cdot 10^{1.5M} . $$This represents the energy released by an earthquake as a function of its magnitude $$ M . To $$find $$ \frac{dE}{dM} $$, we need to differentiate the function $$ E $$ with respect to $$ M . $$Notice that the function involves an exponential term $$ 10^{1.5M} . $$Apply the chain rule for differentiation. The derivative of $$ 10^{1.5M} $$ with respect to $$ M  is$$ $$ 10^{1.5M} \cdot \ln(10) \cdot 1.5 . $$Multiply this by the constant 25,000 to get the derivative: $$ \frac{dE}{dM} = 25,000 \cdot 10^{1.5M} \cdot \ln(10) \cdot 1.5 . $$Now, substitute $$ M = 3 $$ into the derivative $$ \frac{dE}{dM}  to $$evaluate it at this specific magnitude. This involves calculating $$ 25,000 \cdot 10^{4.5} \cdot \ln(10) \cdot 1.5 . $$The derivative $$ \frac{dE}{dM} $$ represents the rate of change of energy with respect to the magnitude of the earthquake. It tells us how much the energy released by the earthquake increases for a small increase in magnitude $$ M . $$The units of this derivative are joules per change in magnitude.

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Key Concepts

Derivative

Exponential Functions

Magnitude Scale