Optimizing Hill Height: A 3D Problem

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SUMMARY

The hill's height is defined by the function h(x, y) = 10(2xy − 3x² − 4y² − 18x + 28y + 12). To find the top of the hill, one must compute the partial derivatives of h with respect to x and y, set them to zero, and solve the resulting system of equations to identify critical points. The maximum height can then be determined by evaluating h at these critical points. For the slope at a specific point (1 mile north and 1 mile east), the gradient must be calculated to find both the steepness and the direction of the steepest slope.

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  • Understanding of partial derivatives
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  • Familiarity with gradient vectors
  • Basic calculus concepts related to optimization
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  • Study the method for finding critical points in optimization problems
  • Explore gradient vectors and their applications in determining slope
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Homework Statement



3. The height of a certain hill (in feet) is given by
h(x, y) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12)
where y is the distance (in miles) north, x is the distance (in miles)
east of the village.
(a) Where is the top of the hill located?
(b) How high is the hill?
(c) How steep is the slope (in feet per mile) at a point 1 mile north
and one mile east of the village? In what direction is the slope
steepest at that point?

Homework Equations





The Attempt at a Solution



I would think that I would have to use the derivative test to get the extrema and go from there, but we didn't cover that material yet. How else can I approach this?
 
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Just take the partial derivatives of h, set them both equal to zero, and solve the system. Then test the points to see which one gives you the largest value for h.

Find the gradient for part (c).
 

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