Question

the intensity of an earthquake can be calculated using the formula

$i = i_0(10)^m$

where $i$ represents the intensity of the earthquake, $m$ is the magnitude of the earthquake on the richter scale, and $i_0$ is the baseline intensity.

on the richter scale, a magnitude 5 earthquake can cause damage to buildings. one of the largest earthquakes ever recorded was in valdivia chile. the earthquake registered a staggering 9.5 on the richter scale. how many times as intense was the valdivia earthquake than a magnitude 5 earthquake? (round your answer to the nearest whole number)

times as intense

Explanation:

Step1: Find intensity of magnitude 5 quake

Use formula \( I = I_0(10)^M \), set \( M = 5 \).
\( I_5 = I_0(10)^5 \)

Step2: Find intensity of Valdivia quake

Set \( M = 9.5 \) in the formula.
\( I_{9.5} = I_0(10)^{9.5} \)

Step3: Find the ratio of intensities

Divide \( I_{9.5} \) by \( I_5 \):
\( \frac{I_{9.5}}{I_5} = \frac{I_0(10)^{9.5}}{I_0(10)^5} \)
Simplify (cancel \( I_0 \), use exponent rule \( \frac{10^a}{10^b} = 10^{a - b} \)):
\( 10^{9.5 - 5} = 10^{4.5} \)

Step4: Calculate \( 10^{4.5} \)

\( 10^{4.5} = 10^{4 + 0.5} = 10^4 \times 10^{0.5} \)
\( 10^4 = 10000 \), \( 10^{0.5} \approx 3.1623 \)
Multiply: \( 10000 \times 3.1623 \approx 31623 \)

Answer:

31623