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Chapter 12: Problem 35

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=\sqrt{25-x^{2}-y^{2}} ;[-6,6] \times[-6,6]$$

Short Answer

Expert verified
Answer: The radii of the level curves for the constants 3 and 4 are 4 and 3, respectively.

Step by step solution

01

Set the function constant and find level curve equations

Choose two constants, say \(k_1\) and \(k_2\). Set the function equal to each constant and solve for the equation of the level curve. $$ z = \sqrt{25-x^2-y^2} = k $$ Squaring both sides gives: $$ x^2+y^2= 25 - k^2 \\\\ $$ For example, let's choose two constants: - \(k_1=3\) $$ x^2+y^2 = 25 - 3^2 = 16 $$ - \(k_2=4\) $$ x^2+y^2 = 25 - 4^2 = 9 $$
02

Graph the cycles and label with z-values

Within the given window, the level curve for \(k_1\) is a circle centered on the origin and has radius 4. Label this curve with its z-value of 3. Similarly, the level curve for \(k_2\) is a circle centered on the origin also with radius 3. Label this curve with its z-value of 4. Note that the domain and range of the actual function are both \([-5, 5]\); hence we only need to draw these cycles with the given window. In conclusion, we have obtained and graphed two level curves, circles with radii 4 and 3. We have labeled them with their respective z-values, 3 and 4, within the window \([-6,6] \times [-6,6]\).

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