The problem provides the formula for the magnitude of an earthquake: $R = \log(\frac{a}{T}) + B$. We are given $R = 6.9$, $B = 3.2$, and $a = 10025$. We need to find the value of $T$.

1. Problem Description

The problem provides the formula for the magnitude of an earthquake: $R = \log(\frac{a}{T}) + B$. We are given $R = 6.9$, $B = 3.2$, and $a = 10025$. We need to find the value of $T$.

2. Solution Steps

We have the equation $R = \log(\frac{a}{T}) + B$. We are given $R=6.9$, $B=3.2$, and $a=10025$. Plugging these values into the equation, we have:

$6.9 = \log(\frac{10025}{T}) + 3.2$

Subtract 3.2 from both sides:

$6.9 - 3.2 = \log(\frac{10025}{T})$

$3.7 = \log(\frac{10025}{T})$

Since the logarithm is base 10, we can rewrite this as:

$10^{3.7} = \frac{10025}{T}$

$10^{3.7} \approx 5011.87$

$5011.87 = \frac{10025}{T}$

Multiply both sides by $T$:

$5011.87 T = 10025$

Divide both sides by 5011.87:

$T = \frac{10025}{5011.87}$

$T \approx 2.0$

3. Final Answer

The length of time that a wave lasts is approximately 2.0 seconds.