Question:** A geographer is studying elevation data and finds that the elevation at Point A is modeled by $ h(x) = 3x^2 - 6x + 5 $, and at Point B by $ k(x) = 2x^2 - 4x + m $. If the elevation at $ x = 2 $ is the same for both points, what is the value of $ m $?
Title: Solving a Quadratic Elevation Model: Finding the Value of m
In the field of geography, understanding elevation changes is crucial for mapping terrain, planning infrastructure, and studying environmental patterns. One common approach involves using mathematical models to represent elevation at specific locations. In this article, we explore a practical scenario involving two elevation functions and determine the value of an unknown parameter, $ m $, based on condition of equality at a given point.
We are given two elevation models:
- At Point A: $ h(x) = 3x^2 - 6x + 5 $
- At Point B: $ k(x) = 2x^2 - 4x + m $
The elevation at $ x = 2 $ is the same for both points. This gives us the opportunity to solve for $ m $.
Step 1: Evaluate $ h(2) $ Substitute $ x = 2 $ into $ h(x) $:
$$ h(2) = 3(2)^2 - 6(2) + 5 = 3(4) - 12 + 5 = 12 - 12 + 5 = 5 $$
So, $ h(2) = 5 $
Step 2: Set $ k(2) $ equal to 5 Now evaluate $ k(2) $ and set it equal to the known elevation at Point A:
$$ k(2) = 2(2)^2 - 4(2) + m = 2(4) - 8 + m = 8 - 8 + m = m $$
Since $ k(2) = h(2) = 5 $, we have:
$$ m = 5 $$
Conclusion: The value of $ m $ that ensures the elevation at $ x = 2 $ is the same for both points is $ oxed{5} $. This demonstrates how algebraic modeling supports accurate geographic analysis and reinforces the importance of verifying parameters in real-world applications.
Keywords: elevation modeling, quadratic functions, geographer, parameter determination, algebra in geography, $ h(x) $, $ k(x) $, $ m $ value, $ x = 2 $, terrain analysis.
