The magnitude of an earthquake is given by [tex]M = log(\frac{I}{S})[/tex] where [tex]I[/tex] is the intensity of the earthquake (measured by the amplitude of the seismograph wave) and [tex]S[/tex] is the intensity of a "standard" earthquake, which is barely detectable. What is the magnitude of an earthquake that is 35 times more intense than a standard earthquake? Use a calculator. Round your answer to the nearest tenth.
Answer (1)
Substitute the given intensity I = 35 S into the magnitude formula M = lo g ( S I ) .
Simplify the expression to M = lo g ( 35 ) .
Calculate the logarithm: M ≈ 1.544 .
Round to the nearest tenth: 1.5 .
Explanation
Understanding the Problem We are given the formula for the magnitude of an earthquake: M = lo g ( S I ) , where I is the intensity of the earthquake and S is the intensity of a standard earthquake. We are also given that the earthquake is 35 times more intense than a standard earthquake, which means I = 35 S . We need to find the magnitude M of this earthquake and round the answer to the nearest tenth.
Substitution Substitute I = 35 S into the formula M = lo g ( S I ) : M = lo g ( S 35 S )
Simplification Simplify the expression: M = lo g ( 35 ) Now, we need to calculate the value of lo g ( 35 ) .
Calculation Using a calculator, we find that lo g ( 35 ) ≈ 1.544068 .
Rounding Round the result to the nearest tenth: M ≈ 1.5
Conclusion Therefore, the magnitude of the earthquake is approximately 1.5.
Examples
Earthquakes release energy that can be measured using the Richter scale, which is a logarithmic scale. This scale helps scientists and engineers assess the severity of earthquakes and design structures that can withstand seismic activity. For example, understanding the magnitude of an earthquake helps in constructing buildings that can resist the forces generated during such events, ensuring public safety.
