5Question: An ichthyologist models the growth of a fish population with the quadratic function $ f(x) = x^2 - 4x + m $. If the population at $ x = 3 $ is 7, find $ m $. - Silent Sales Machine
Title: Discovering the Secret Constant: Solving for m in the Fish Population Model
Title: Discovering the Secret Constant: Solving for m in the Fish Population Model
In the delicate balance of aquatic ecosystems, understanding fish population dynamics is crucial for conservation and management. One powerful tool in this endeavor is mathematical modeling. Today, we explore a real-world example where an ichthyologist uses a quadratic function to predict fish populations over time.
Consider the growth model:
$$ f(x) = x^2 - 4x + m $$
This function describes the population of a fish species over time, with $ x $ representing time (in months or seasons) and $ m $ being a critical parameter influenced by environmental conditions.
Understanding the Context
An essential step in validating this model is verifying it against observed data points. Suppose measurements show that at $ x = 3 $ months, the actual population is 7. We can use this data to determine the unknown constant $ m $.
Step-by-Step: Solving for $ m $
Start with the given function evaluated at $ x = 3 $:
$$
f(3) = (3)^2 - 4(3) + m
$$
Calculate the expression step-by-step:
$$
f(3) = 9 - 12 + m = -3 + m
$$
Key Insights
We know from field data that $ f(3) = 7 $. Therefore:
$$
-3 + m = 7
$$
Solve for $ m $:
$$
m = 7 + 3 = 10
$$
Conclusion
The value of $ m $, representing a key environmental factor in this model, is $ oxed{10} $. This precise calculation ensures the population model accurately reflects observed fish numbers at critical time points.
By linking mathematical functions to ecological data, ichthyologists refine their predictions, support sustainable fishing practices, and contribute to the long-term health of aquatic populations. The simplicity of a quadratic hidden behind complex ecology reminds us: behind every species lies a story shaped by numbers—and calculated with care.
