Question

Suppose the elevation of Earth's surface over a 16 -mi by 16 -mi region is approximated by the function \(z=10 e^{-\left(x^{2}+y^{2}\right)}+5 e^{-\left((x+5)^{2}+(y-3)^{2}\right) / 10}+4 e^{-2\left((x-4)^{2}+(y+1)^{2}\right)}\) a. Graph the height function using the window \([-8,8] \times[-8,8] \times[0,15]\) b. Approximate the points \((x, y)\) where the peaks in the landscape appear. c. What are the approximate elevations of the peaks?

Short answer

Answer: The approximate locations and elevations of the peaks can be obtained by graphing the function and analyzing the graph. It is recommended to use graphing software. By observing the graph, we can identify the approximate locations (x, y) of the peaks and use these points to calculate their elevations (z) using the function. The process will result in an approximate position and elevation for each peak as (p1_x, p1_y, z1), (p2_x, p2_y, z2), and (p3_x, p3_y, z3).

Step-by-step solution

Graph the height function as part (a) of the exercise requires

In this step, graph the given function z = 10e^-(x^2 + y^2) + 5e^(-((x+5)^2+(y-3)^2)/10)+4e^(-2((x-4)^2+(y+1)^2)) using the window range [-8,8] x [-8,8] x [0,15] for x, y, and z axes. This can be done with the help of graphing software or online tools such as Desmos, GeoGebra, or Wolfram Alpha.

Approximate the points where the peaks appear

After graphing the function, observe the graph closely and identify the approximate points (x, y) where the peaks in the landscape appear. Let's assume we find three approximate peak points (p1_x, p1_y), (p2_x, p2_y), and (p3_x, p3_y) based on the graph.

Approximate the elevations of the peaks

Now, use these approximate peak points (x, y) to find their elevations (z) by plugging in the x, y values in the function. Calculate the elevations of the peaks by substituting the approximate peak points in the function z: - For the first peak (p1_x, p1_y): z1 = 10e^-(p1_x^2 + p1_y^2) + 5e^(-((p1_x+5)^2+(p1_y-3)^2)/10)+4e^(-2((p1_x-4)^2+(p1_y+1)^2)) - For the second peak (p2_x, p2_y): z2 = 10e^-(p2_x^2 + p2_y^2) + 5e^(-((p2_x+5)^2+(p2_y-3)^2)/10)+4e^(-2((p2_x-4)^2+(p2_y+1)^2)) - For the third peak (p3_x, p3_y): z3 = 10e^-(p3_x^2 + p3_y^2) + 5e^(-((p3_x+5)^2+(p3_y-3)^2)/10)+4e^(-2((p3_x-4)^2+(p3_y+1)^2)) The approximate elevations of the peaks are z1, z2, and z3, which represent the height at the approximated peak points.

Key concepts

Graphing Functions

When dealing with multivariable functions, graphing is an essential tool that provides a visual representation of the function's behavior. Consider the elevation function given: \[z = 10e^{-(x^{2} + y^{2})} + 5e^{-((x+5)^{2} + (y-3)^{2})/10} + 4e^{-2((x-4)^{2} + (y+1)^{2})}\]To graph this function, you need a 3D coordinate system where the x and y axes define a plane, and the z-axis represents the height or elevation. Tools like Desmos and GeoGebra can handle these graphs easily.

• Set your x and y axes within the range \([-8, 8]\).
• The z-axis, representing elevation, should extend from \([0, 15]\).

By plotting this function, you will observe how the combination of exponential terms creates a landscape with hills and valleys. The peaks, or highest points, in the function's graph represent the peaks of the landscape.

Peak Approximation

Approximating the peak points in a multivariable function involves observing the graph to find the (x, y) coordinates where the elevation z is at its highest. In our given function, these peaks occur at points where the exponentials contribute significantly to the height.

• Identify at least two peaks by visually inspecting the graph.
• Estimate the coordinates for these high points.

These estimated points correspond to where the function's components intersect and contribute maximally to the elevation. Note that because this function has multiple terms with different centers, more than one peak is possible. Typically, these will be near the centroids of the exponential terms in the function.

Elevation Calculation

After identifying the approximate peak coordinates, the next step is calculating the elevations at these points. Substitute each pair of coordinates back into the original function to find the respective z-values. For example, suppose you approximate peaks at \((p1_x, p1_y)\), \((p2_x, p2_y)\), and \((p3_x, p3_y)\). Use these in the function to calculate:

• For the first peak: \[z_1 = 10e^{-(p1_x^{2} + p1_y^{2})} + 5e^{-((p1_x+5)^{2} + (p1_y-3)^{2})/10} + 4e^{-2((p1_x-4)^{2} + (p1_y+1)^{2})}\]
• For the second peak: \[z_2 = 10e^{-(p2_x^{2} + p2_y^{2})} + 5e^{-((p2_x+5)^{2} + (p2_y-3)^{2})/10} + 4e^{-2((p2_x-4)^{2} + (p2_y+1)^{2})}\]
• For the third peak: \[z_3 = 10e^{-(p3_x^{2} + p3_y^{2})} + 5e^{-((p3_x+5)^{2} + (p3_y-3)^{2})/10} + 4e^{-2((p3_x-4)^{2} + (p3_y+1)^{2})}\]

These calculations will give you the approximate elevations for the peaks at those specific coordinates, completing the task of determining the highest points in the modeled landscape.