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Chapter 13: Q 36 (page 1066)

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.
Change $蠒 = \frac{蟺}{4}$ to the rectangular and spherical systems.

Short Answer

Expert verified

The equation in rectangular coordinates is $x^{2} + y^{2} - z^{2} = 0$

Step by step solution

01

Step 1:Given information

The given expression is $蠒 = \frac{蟺}{4}$

02

Simplification (a)

To convert $蠒 = \frac{蟺}{4}$ into rectangular coordinates

We have

$r = \sqrt{x^{2} + y^{2} + z^{2}}, \tan 胃 = \frac{y}{x}, \cos 蠒 = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

$鈬� \cos 蠒 = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

$鈬� 蠒 = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Substituting $鈬� 蠒 = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$ in $蠒 = \frac{蟺}{4}$

$鈬� \frac{蟺}{4} = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

$鈬� \cos \frac{蟺}{4} = (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

$鈬� (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) = \frac{1}{\sqrt{2}}$

$鈬� (\frac{z^{2}}{x^{2} + y^{2} + z^{2}}) = \frac{1}{2}$

$鈬� x^{2} + y^{2} + z^{2} = 2 z^{2}$

$鈬� x^{2} + y^{2} - z^{2} = 0$

Hence,The equation in rectangular coordinates is $x^{2} + y^{2} - z^{2} = 0$

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Most popular questions from this chapter

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${鈭�}_{c}^{d} {鈭�}_{a}^{b} f (x, y) d x d y$ .

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