Worksheet: Cylindrical and Spherical Coordinates

In this worksheet, we will practice identifying cylindrical and spherical coordinates and converting points and equations between Cartesian, cylindrical, and spherical coordinates.

Q1:

Write the given equation 𝑥+𝑦+9𝑧=36 in cylindrical and spherical coordinates.

  • A 𝜌 + 9 𝑧 = 3 6 , 𝑟 1 + 8 𝜙 = 3 6 c o s
  • B 𝜌 + 9 𝑧 = 3 6 , 𝑟 1 + 8 𝜙 = 3 6 c o s
  • C 𝜌 + 9 𝑧 = 3 6 , 𝑟 = 3 6
  • D 𝜌 + 9 𝑧 = 3 6 , 𝑟 1 + 8 𝜙 = 3 6 s i n
  • E 𝜌 + 9 𝑧 = 3 6 , 𝑟 = 3 6

Q2:

Convert the point 2,23,1 from Cartesian coordinates to cylindrical and spherical coordinates, rounding the value of 𝜙 to two decimal places.

  • A 4 , 𝜋 3 , 1 , 1 7 , 𝜋 3 , 0 . 5 7
  • B 1 7 , 2 𝜋 3 , 1 , 1 7 , 2 𝜋 3 , 1 . 8 2
  • C 4 , 2 𝜋 3 , 1 , 1 7 , 2 𝜋 3 , 0 . 5 7
  • D 4 , 𝜋 3 , 1 , 1 7 , 𝜋 3 , 1 . 8 2
  • E 4 , 𝜋 6 , 1 , 1 7 , 𝜋 6 , 1 . 8 2

Q3:

Express the equation 𝑥+𝑦=2𝑦 in cylindrical and spherical coordinates.

  • A 𝜌 = 2 𝜃 s i n , 𝑟 𝜙 = 2 𝜃 s i n s i n
  • B 𝜌 = 𝜃 c o s , 𝑟 𝜙 = 𝜃 s i n c o s
  • C 𝜌 = 2 𝜃 c o s , 𝑟 𝜙 = 2 𝜃 s i n c o s
  • D 𝜌 = 2 𝜃 s i n , 𝑟 𝜙 = 2 𝜃 s i n s i n
  • E 𝜌 = 𝜃 s i n , 𝑟 𝜙 = 𝜃 s i n s i n

Q4:

Let 𝑃=(𝑎,𝜃,𝜙) be a point in spherical coordinates with 𝑎>0 and 0<𝜙<𝜋, where 𝑃 lies on the sphere 𝜌=𝑎. Since 0<𝜙<𝜋, the line segment from the origin to 𝑃 can be extended to intersect the cylinder given by 𝑟=𝑎 in cylindrical coordinates. Find the cylindrical coordinates of that point of intersection.

  • A ( 0 , 𝜃 , 𝑎 𝜙 ) c o s
  • B ( 0 , 𝜃 , 𝑎 𝜙 ) c o t
  • C ( 𝑎 , 𝜃 , 𝑎 𝜙 ) c o t
  • D ( 𝑎 , 𝜃 , 𝑎 𝜙 ) c o s
  • E ( 𝑎 , 𝜙 , 𝑎 𝜃 ) c o t

Q5:

Convert the point 21,7,0 from Cartesian coordinates to cylindrical and spherical coordinates.

  • A 2 7 , 1 1 𝜋 6 , 0 , 2 7 , 1 1 𝜋 6 , 2 𝜋 3
  • B 7 1 0 , 𝜋 6 , 0 , 7 1 0 , 𝜋 6 , 2 𝜋 3
  • C 7 1 0 , 1 1 𝜋 6 , 0 , 7 1 0 , 1 1 𝜋 6 , 𝜋 2
  • D 2 7 , 1 1 𝜋 6 , 0 , 2 7 , 1 1 𝜋 6 , 𝜋 2
  • E 2 7 , 𝜋 6 , 0 , 2 7 , 𝜋 6 , 𝜋 2

Q6:

Write the given equation 𝑥+𝑦+𝑧=25 in cylindrical and spherical coordinates.

  • A 𝑧 𝜌 = 2 5 , 𝑟 = 2 5
  • B 𝑧 + 𝜌 = 2 5 , 𝑟 = 5
  • C 𝑧 + 𝜌 = 2 5 , 𝑟 = 5
  • D 𝑧 + 𝜌 = 2 5 , 𝑟 = 2 5
  • E 𝑧 𝜌 = 2 5 , 𝑟 = 5

Q7:

Which of the following is NOT a correct description of the sphere of radius 𝑎 in 𝟛 centered at the origin?

  • A ( 𝑟 , 𝜃 , 𝑧 ) 𝑧 = 𝑎 𝑟 :
  • B ( 𝑥 , 𝑦 , 𝑧 ) 𝑥 + 𝑦 = 𝑎 𝑧 :
  • C ( 𝑟 , 𝜃 , 𝑧 ) 𝑧 = 𝑎 𝑟 :
  • D { ( 𝑟 , 𝜃 , 𝜙 ) 𝑟 = 𝑎 } :

Q8:

Convert the point 0,2,2 from Cartesian coordinates to cylindrical, (𝑟,𝜃,𝑧), and spherical, (𝜌,𝜃,𝜙), coordinates. Round the value of 𝜙 to two decimal places.

  • A 2 , 𝜋 2 , 2 , 2 2 , 𝜋 2 , 0 . 6 8
  • B 2 , 𝜋 , 2 , 6 , 𝜋 , 0 . 6 8
  • C 2 , 3 𝜋 2 , 2 , 6 , 3 𝜋 2 , 0 . 6 8
  • D 2 , 𝜋 2 , 2 , 6 , 𝜋 2 , 0 . 6 2
  • E ( 2 , 𝜋 , 2 ) , 2 2 , 𝜋 , 0 . 6 2

Q9:

Convert the point (5,5,6) from Cartesian coordinates to cylindrical, (𝑟,𝜃,𝑧), and spherical, (𝜌,𝜃,𝜙), coordinates. Round the value of 𝜙 to two decimal places.

  • A 5 2 , 𝜋 4 , 6 , 5 2 , 𝜋 4 , 1 . 1 8
  • B 8 6 , 3 𝜋 4 , 6 , 8 6 , 3 𝜋 4 , 5 4 . 7 3
  • C 5 2 , 3 𝜋 4 , 6 , 8 6 , 3 𝜋 4 , 4 9 . 6 8
  • D 5 , 𝜋 3 , 6 , 5 2 , 𝜋 3 , 3 0 . 0 0
  • E 8 6 , 𝜋 4 , 6 , 8 6 , 𝜋 4 , 5 2 6

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