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The surface of a mountain is modeled by the equation h(x, y) = 7000 − 0.001x2 − 0.004y2. A mountain climber is at the point (700, 500, 5510). In what direction should the climber move in order to ascend at the greatest rate?



Answer :

To ascend the mountain at the greatest rate, the climber should move in the direction (-0.33, -0.94).

To find the direction in which the climber should move to ascend the mountain at the greatest rate, we need to find the gradient of the height function h(x, y) = 7000 − 0.001x^2 − 0.004y^2.

The gradient vector (∇h) is calculated by finding the partial derivatives of h with respect to x and y:

  • ∂h/∂x = -0.002x
  • ∂h/∂y = -0.008y

At the point (700, 500), the partial derivatives are:

  • ∂h/∂x = -0.002 * 700 = -1.4
  • ∂h/∂y = -0.008 * 500 = -4

Thus, the gradient vector at (700, 500) is ∇h = (-1.4, -4).

Since the gradient vector points in the direction of the steepest ascent, the climber should move in the direction of the vector (-1.4, -4).

However, to obtain a unit vector in this direction, we normalize the gradient vector:

|∇h| = √( (-1.4)^2 + (-4)^2 ) = √( 1.96 + 16 ) = √17.96 ≈ 4.24

The unit vector is:

u = 1/4.24 * (-1.4, -4) ≈ (-0.33, -0.94)

Therefore, the climber should move in the direction approximately given by (-0.33, -0.94) to ascend the mountain at the greatest rate.

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