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Chapter 16: Problem 23

The cylindrical coordinates \(\left(5,30^{\circ}, 12\right)\) are expressed in spherical coordinates as a) \(\left(13,30^{\circ}, 67.4^{\circ}\right)\) c) \(\left(15,52.6^{\circ}, 22.6^{\circ}\right)\) b) \(\left(13,30^{\circ}, 22.6^{\circ}\right)\) d) \(\left(15,52.6^{\circ},-22.6^{\circ}\right)\)

Short Answer

Expert verified
Question: Convert the given cylindrical coordinates \((\rho, \phi, z) = (5, 30°, 12)\) to spherical coordinates. Answer: The equivalent spherical coordinates for the given cylindrical coordinates are \((r, \theta, \phi) = (13, 22.6^{\circ}, 30^{\circ})\).

Step by step solution

01

Calculate r

Using the given cylindrical coordinates (\(\rho = 5\), \(z = 12\)), we can compute the value of \(r\) using the formula \(r = \sqrt{\rho^2 + z^2}\). So we have, \(r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).
02

Calculate theta

Next, we compute the value of \(\theta\) using the formula \(\theta = \arctan(\frac{\rho}{z})\). So we have, \(\theta = \arctan(\frac{5}{12}) = 22.6^{\circ}\).
03

Determine phi

The given cylindrical coordinates have \(\phi = 30^{\circ}\), and this value remains the same in the spherical coordinates system.
04

Compare the results with the options

Now we need to compare our computed values of the spherical coordinates to see which set matches our values. Our computed spherical coordinates values are \((r, \theta, \phi) = (13, 22.6^{\circ}, 30^{\circ})\). Comparing our results, the correct spherical coordinates are given in option (b): \(\left(13,30^{\circ}, 22.6^{\circ}\right)\).

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