According to the US Geological Survey, the number of kilograms of TNT you would need to detonate to release the same amount of energy as an earthquake is given by the formula below, in which M is the earthquake magnitude.

If an earthquake released four billion kg of TNT worth of energy, what magnitude was it? Round your answer to two digits after the decimal.


The formula used to determine the equivalent energy in kilograms of TNT for an earthquake is:

\[
E = 10^{\left( \frac{3M}{2} - 7 \right)}
\]

where:
- \( E \) is the energy released in kilograms of TNT, and
- \( M \) is the earthquake magnitude.

In this case, we are given \( E = 4,000,000,000 \) kg of TNT, and we need to solve for \( M \).

### Steps to solve:

1. Substitute the given value of \( E \) into the formula:
\[
4,000,000,000 = 10^{\left( \frac{3M}{2} - 7 \right)}
\]

2. Take the logarithm (base 10) of both sides:
\[
\log_{10}(4,000,000,000) = \frac{3M}{2} - 7
\]

3. Solve for \( M \).

I'll compute this now.The earthquake that released the equivalent of four billion kilograms of TNT had a magnitude of approximately \( 11.07 \).

Would you like further details on the calculation?

Here are five related questions:
1. How does the magnitude scale relate to energy release?
2. What is the highest recorded earthquake magnitude?
3. How accurate is this formula for large magnitudes?
4. Can earthquake magnitude decrease after the initial shock?
5. What are other ways to measure earthquake intensity?

**Tip:** The Richter scale is logarithmic, meaning each whole number increase represents a tenfold increase in amplitude and roughly 31.6 times more energy.

If an earthquake released four billion kg of TNT worth of energy, what magnitude was it? Round your answer to two digits after the decimal.

Discover how to calculate the magnitude of an earthquake based on the energy equivalent to four billion kg of TNT. Learn key mathematical concepts like logarithms and exponential equations, with a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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