Worksheet: Cylindrical and Spherical Coordinates

In this worksheet, we will practice using cylindrical and spherical coordinates to study of three-dimensional phenomena for some models.

Q1:

Write the given equation 𝑥 + 𝑦 + 9 𝑧 = 3 6 2 2 2 in cylindrical and spherical coordinates.

  • A 𝜌 + 9 𝑧 = 3 6 2 2 , 𝑟 = 3 6 2
  • B 𝜌 + 9 𝑧 = 3 6 2 , 𝑟 1 + 8 𝜙 = 3 6 c o s 2
  • C 𝜌 + 9 𝑧 = 3 6 2 2 , 𝑟 1 + 8 𝜙 = 3 6 2 2 s i n
  • D 𝜌 + 9 𝑧 = 3 6 2 2 , 𝑟 1 + 8 𝜙 = 3 6 2 2 c o s
  • E 𝜌 + 9 𝑧 = 3 6 2 , 𝑟 = 3 6 2

Q2:

Convert the point 2 , 2 3 , 1 from Cartesian coordinates to cylindrical and spherical coordinates, rounding the value of 𝜙 to two decimal places.

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

Q3:

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

  • A 𝜌 = 𝜃 s i n , 𝑟 𝜙 = 𝜃 s i n s i n
  • B 𝜌 = 2 𝜃 c o s , 𝑟 𝜙 = 2 𝜃 s i n c o s
  • C 𝜌 = 𝜃 c o s , 𝑟 𝜙 = 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 = 2 𝜃 2 s i n , 𝑟 𝜙 = 2 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 t
  • B ( 𝑎 , 𝜃 , 𝑎 𝜙 ) c o s
  • C ( 0 , 𝜃 , 𝑎 𝜙 ) c o s
  • D ( 𝑎 , 𝜃 , 𝑎 𝜙 ) c o t
  • E ( 𝑎 , 𝜙 , 𝑎 𝜃 ) c o t

Q5:

Convert the point 2 1 , 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 2 7 , 𝜋 6 , 0 , 2 7 , 𝜋 6 , 𝜋 2
  • 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 7 1 0 , 𝜋 6 , 0 , 7 1 0 , 𝜋 6 , 2 𝜋 3

Q6:

Write the given equation 𝑥 + 𝑦 + 𝑧 = 2 5 2 2 2 in cylindrical and spherical coordinates.

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

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