Question

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ z=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4,5 $$

Short answer

The level curves are six concentric circles with centres at the origin and decreasing radii from \(\sqrt{25}\) to \(\sqrt{0}\) as the \(c\) value increases.

Step-by-step solution

Solve for x and y

In this step, we want to solve the equation \(z = \sqrt{25 - x^2 - y^2} = c\) for \(x\) and \(y\). By squaring both sides of the equation we remove the square root from the right side: \(z^2 = 25 - x^2 - y^2\) or \(x^2 + y^2 = 25 - z^2\).

Use the constant values

Now we can substitute the given constants for \(c = 0, 1, 2, 3, 4, 5\) into the equation obtained from step 1. We get the following set of equations: \[\] 1) When \(c = 0\), \(x^2 + y^2 = 25\); \[\] 2) When \(c = 1\), \(x^2 + y^2 = 24\); \[\] 3) When \(c = 2\), \(x^2 + y^2 = 21\); \[\] 4) When \(c = 3\), \(x^2 + y^2 = 16\); \[\] 5) When \(c = 4\), \(x^2 + y^2 = 9\); \[\] 6) When \(c = 5\), \(x^2 + y^2 = 0\).

Sketch the level curves

Each of the equations found in step 2 correspond to circles centred on the origin with radii of \(\sqrt{25}, \sqrt{24}, \sqrt{21}, \sqrt{16}, \sqrt{9}\) and \(\sqrt{0}\) respectively. Now the task is to draw these circles in the x-y plane. The resulting graph would have 6 concentric circles with decreasing radii.