Final answer:
To reach the top of the hill fastest, you should proceed in the direction that maximizes the rate of increase of the hill's height. In this case, you should initially proceed in the direction (-1,-1,0). The angle above the horizontal at which you will be climbing initially is π/2 radians.
Step-by-step explanation:
To reach the top of the hill fastest, you should proceed in the direction that maximizes the rate of increase of the hill's height. In other words, you need to find the direction in which the gradient of the hill is steepest.
The gradient of the hill is given by the partial derivatives of the height function with respect to the x and y coordinates. Evaluating these partial derivatives at the point (70,90,300) gives:
∂z/∂x = -0.05x = -0.05 * 70 = -3.5
∂z/∂y = -0.07y = -0.07 * 90 = -6.3
Therefore, the steepest direction is in the negative x and negative y directions, which means that you should proceed initially in the direction (-1,-1,0).
To find the angle above the horizontal at which you will be climbing initially, you can use the dot product between the direction vector and the unit vector pointing upwards (0,0,1). The dot product is given by:
cos(θ) = (a · b) / (|a| * |b|)
Where a is the direction vector (-1,-1,0) and b is the unit vector (0,0,1). Evaluating the dot product gives:
cos(θ) = (0 * -1 + 0 * -1 + 1 * 0) / (sqrt(1) * sqrt(2)) = 0
Since the dot product is 0, the angle θ is 90 degrees or π/2 radians. Therefore, you will be climbing initially at an angle of π/2 radians above the horizontal.