Question

In Exercises \(41-46,\) convert the point from cylindrical coordinates to spherical coordinates. $$ (12, \pi, 5) $$

Short answer

The spherical coordinates of the given point in cylindrical coordinates (12, \(\pi\), 5) are (13, 1.176, \(\pi\)).

Step-by-step solution

Identify the cylindrical coordinates

The given point is (12, \(\pi\), 5) in cylindrical coordinates, with radial distance \(r = 12\), angle \(\theta = \pi\), and height \(z = 5\).

Compute rho

Calculate \(\rho\) using the relationship \(\rho = \sqrt{r^2 + z^2}\). Substitute the given values into the equation to yield:\[\rho = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.\]

Compute \(\phi\)

Next, calculate \(\phi\) using the relationship \(\phi = \arctan{(r/z)}\). Plug in the given values to get:\[\phi = \arctan{(12/5)} = 1.176.\]

Identify \(\psi\)

\(\psi = \theta\) in the context of converting cylindrical coordinates into spherical coordinates. Thus, we have \(\psi = \theta = \pi\).