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.
