Question

The surface of a mountain is modeled by the equation \(h(x, y)=5000-0.001 x^{2}-0.004 y^{2}\). A mountain climber is at the point (500,300,4390) . In what direction should the climber move in order to ascend at the greatest rate?

Short answer

The climber should move in the direction of the gradient vector of \(h(x, y)\) at the point (500,300), which is \(\langle -1,-2.4 \rangle\), or in the direction of its corresponding unit vector for the path of steepest ascent.

Step-by-step solution

Find the Gradient of \(h(x, y)\)

The gradient of a function \(h(x, y)\) is a vector that points in the direction of the function's greatest rate of increase. It is given by the vector of the function's first partial derivatives respect to x and y, denoted as \(\nabla h = \langle h_x, h_y \rangle\). Apply these rules to find the partial derivatives of \(h(x, y)=5000-0.001x^{2}-0.004y^{2}\). Find the derivative of \(h(x, y)\) with respect to \(x\) to get \(h_x\): \(h_x=-0.002x\). Then find the derivative of \(h(x, y)\) with respect to \(y\) to get \(h_y\): \(h_y=-0.008y\). So, \(\nabla h = \langle -0.002x,-0.008y \rangle\).

Evaluate the Gradient at the Climber's Current Position

The climber is currently at the point (500,300,4390). Evaluating the gradient at this point gives us the direction the climber needs to move for the greatest rate of ascent. Plugging these values into \(\nabla h\), we get: \(-0.002(500)\) for \(h_x\) and \(-0.008(300)\) for \(h_y\), which yields \(\nabla h = \langle -1,-2.4 \rangle\) at this point.

Calculate the Unit Vector in the Direction of Steepest Ascent

While \(\nabla h\) gives us the direction of steepest ascent, we often want the direction as a unit vector, which has a length of 1. The unit vector \(\hat{n}\) in the direction of \(\nabla h\) is given by \(\hat{n} = \frac{\nabla h}{ |\nabla h | }\). Here, \( |\nabla h | \) is the magnitude of the gradient vector, calculated using the Pythagorean theorem: \(\sqrt{(-1)^2 + (-2.4)^2}\). After finding the magnitude, divide the components of \(\nabla h\) by it to find the unit vector.

Interpret the Result

Based on these calculations, the climber should move in the direction of the gradient vector, which is the direction in which \(h(x, y)\) increases most quickly. This is the direction of the unit vector obtained from the previous step.

Key concepts

Gradient Vector

Imagine standing on a vast mountain terrain, looking for the highest point. The gradient vector is like a compass that points towards the peak, guiding you to ascend at the steepest possible path. In calculus, the gradient vector of a function represents the direction and rate of the fastest increase. Mathematically, it's the vector of all partial derivatives of that function. For the given mountain surface described by the equation h(x, y), finding the gradient involves calculating the derivatives with respect to both coordinates, x and y.

For h(x, y) = 5000 - 0.001x^2 - 0.004y^2, the gradient vector is \(abla h = \langle -0.002x, -0.008y \rangle\). This vector provides not just the path of your ascent but also how steep the climb will be; the steeper the slope, the greater the magnitudes of the vector's components.

Partial Derivatives

To fully understand the direction the mountain climber must take, we delve into the partial derivatives of the mountain's equation. These derivatives measure how the height of the mountain surface changes as the climber moves in one direction while keeping the position in the other direction constant.

In this scenario, the partial derivatives are calculated as \(h_x=-0.002x\) and \(h_y=-0.008y\). Respectively, they represent the slope of the terrain if the climber moves directly east (along the x-axis) or directly north (along the y-axis). When adopting this gradient approach, we can visualize the climber examining a detailed map, where these partial derivatives represent the steepness of the incline in two perpendicular directions.

Steepest Ascent

Finding the path of steepest ascent is crucial for our mountain climber to reach greater heights most efficiently. Steepest ascent refers to the direction in which the increase in altitude is the fastest. It is the path where each step taken results in the maximum possible elevation gain.

As we calculated before, the direction of steepest ascent for the function h(x, y) at a given point is the direction of the gradient vector. For the climber at coordinates (500, 300), the gradient vector \(abla h \) is \(\langle -1, -2.4 \rangle\). This indicates that to ascend most rapidly, the climber should head in a straight line opposite to the gradient vector's direction because the gradient points uphill and our function is for height reduction.

Unit Vector

The concept of a unit vector is like finding the true north on a compass without the scaling factor—it solely indicates direction. A unit vector has a length of exactly 1, thereby providing directionality without magnitude. In our mountainous terrain, while the gradient vector gives direction, it's the unit vector that simplifies this direction to its essence, stripping away how steep the path is.

To obtain the unit vector in the direction of the gradient, we calculate the magnitude of the gradient vector \(|abla h |\) and then divide each component of the gradient vector by this magnitude. This will give us a vector that points in the same direction as the gradient but has been normalized to a length of 1, making it a true unit vector.