Question: A seismologist models the time between two seismic events as a function $ T(x) = x^2 - 6x + 13 $, where $ x $ is the number of days since monitoring began. What is the minimum time between events predicted by this model? - go-checkin.com

Understanding the Context

The function $ T(x) = x^2 - 6x + 13 $ is a quadratic equation in standard form $ ax^2 + bx + c $, with $ a = 1 $, $ b = -6 $, and $ c = 13 $. Since the coefficient of $ x^2 $ is positive, the parabola opens upwards, meaning it has a minimum value at its vertex.

Finding the Vertex

The x-coordinate of the vertex of a parabola given by $ ax^2 + bx + c $ is calculated using the formula:

$$
x = -rac{b}{2a}
$$

Key Insights

Substituting $ a = 1 $ and $ b = -6 $:

$$
x = -rac{-6}{2(1)} = rac{6}{2} = 3
$$

This means the minimum time between seismic events occurs 3 days after monitoring began.

Calculating the Minimum Time

To find the minimum value of $ T(x) $, substitute $ x = 3 $ into the original function:

Final Thoughts

$$
T(3) = (3)^2 - 6(3) + 13 = 9 - 18 + 13 = 4
$$

Thus, the minimum time predicted by the model is 4 days between seismic events.

Seismological Significance

This result suggests that, despite fluctuations, the system stabilizes over time, with the shortest observed interval between events being 4 days. This insight helps seismologists assess rupture characteristics and recurrence patterns in tectonic activity. Monitoring for repeated intervals near this value may aid in forecasting future events and improving early-warning systems.

Conclusion

The quadratic model $ T(x) = x^2 - 6x + 13 $ predicts a minimum time of 4 days between seismic events, occurring at $ x = 3 $. Recognizing such patterns enables more accurate seismic hazard assessments and enhances preparedness strategies in monitored regions.