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Chapter 16: Problem 24

The equation of a 4 -m-radius sphere using cylindrical coordinates is a) \(x^{2}+y^{2}+z^{2}=16\) c) \(r^{2}+z^{2}=16\) b) \(r^{2}=16\) d) \(x^{2}+y^{2}=16\)

Short Answer

Expert verified
a) r^2 + z^2 = 8 b) r^2 + z^2 = 12 c) r^2 + z^2 = 16 d) r^2 + z^2 = 20

Step by step solution

01

Write the equation of the sphere in Cartesian coordinates

Given the radius of the sphere is 4 meters, we can write the equation in Cartesian coordinates as: \(x^{2} + y^{2} + z^{2} = 4^2\) which simplifies to: \(x^{2} + y^{2} + z^{2} = 16\)
02

Convert the equation to cylindrical coordinates

Now, substitute x = r * cos(θ) and y = r * sin(θ) into the equation: \((r * cos(θ))^{2} + (r * sin(θ))^{2} + z^{2} = 16\)
03

Simplify the equation

Using the trigonometric identity \(cos^2(θ) + sin^2(θ) = 1\), we can simplify the equation as follows: \(r^{2} * (cos^{2}(θ) + sin^{2}(θ)) + z^{2} = 16\) \(r^{2} * 1 + z^{2} = 16\) \(r^{2} + z^{2} = 16\) The correct answer is c) \(r^{2} + z^{2} = 16\).

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