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Lesson: Cylindrical and Spherical Coordinates

Worksheet • 6 Questions

Q1:

Write the given equation π‘₯ + 𝑦 + 9 𝑧 = 3 6 2 2 2 in cylindrical and spherical coordinates.

  • A 𝜌 + 9 𝑧 = 3 6 2 2 , π‘Ÿ ο€Ή 1 + 8 πœ™  = 3 6 2 2 c o s
  • B 𝜌 + 9 𝑧 = 3 6 2 , π‘Ÿ = 3 6 2
  • C 𝜌 + 9 𝑧 = 3 6 2 , π‘Ÿ ο€Ή 1 + 8 πœ™  = 3 6 c o s 2
  • D 𝜌 + 9 𝑧 = 3 6 2 2 , π‘Ÿ ο€Ί 1 + 8 πœ™  = 3 6 2 2 s i n
  • E 𝜌 + 9 𝑧 = 3 6 2 2 , π‘Ÿ = 3 6 2

Q2:

Convert the point ο€» 2 , 2 √ 3 , βˆ’ 1  from Cartesian coordinates to cylindrical and spherical coordinates, rounding the value of πœ™ to two decimal places.

  • A ο€» 4 , πœ‹ 3 , βˆ’ 1  , ο€» √ 1 7 , πœ‹ 3 , 1 . 8 2 
  • B ο€Ό 4 , 2 πœ‹ 3 , βˆ’ 1  , ο€Ό √ 1 7 , 2 πœ‹ 3 , 0 . 5 7 
  • C ο€» 4 , πœ‹ 6 , βˆ’ 1  , ο€» √ 1 7 , πœ‹ 6 , 1 . 8 2 
  • D ο€» 4 , πœ‹ 3 , βˆ’ 1  , ο€» √ 1 7 , πœ‹ 3 , 0 . 5 7 
  • E ο€Ό √ 1 7 , 2 πœ‹ 3 , βˆ’ 1  , ο€Ό √ 1 7 , 2 πœ‹ 3 , 1 . 8 2 

Q3:

Express the equation π‘₯ + 𝑦 = 2 𝑦 2 2 in cylindrical and spherical coordinates.

  • A 𝜌 = 2 πœƒ s i n , π‘Ÿ πœ™ = 2 πœƒ s i n s i n
  • B 𝜌 = 2 πœƒ 2 s i n , π‘Ÿ πœ™ = 2 πœƒ s i n s i n
  • C 𝜌 = 2 πœƒ c o s , π‘Ÿ πœ™ = 2 πœƒ s i n c o s
  • D 𝜌 = πœƒ c o s , π‘Ÿ πœ™ = πœƒ s i n c o s
  • E 𝜌 = πœƒ s i n , π‘Ÿ πœ™ = πœƒ s i n s i n

Q4:

Let 𝑃 = ( π‘Ž , πœƒ , πœ™ ) be a point in spherical coordinates with π‘Ž > 0 and 0 < πœ™ < πœ‹ , where 𝑃 lies on the sphere 𝜌 = π‘Ž . Since 0 < πœ™ < πœ‹ , the line segment from the origin to 𝑃 can be extended to intersect the cylinder given by π‘Ÿ = π‘Ž in cylindrical coordinates. Find the cylindrical coordinates of that point of intersection.

  • A ( π‘Ž , πœƒ , π‘Ž πœ™ ) c o t
  • B ( π‘Ž , πœ™ , π‘Ž πœƒ ) c o t
  • C ( π‘Ž , πœƒ , π‘Ž πœ™ ) c o s
  • D ( 0 , πœƒ , π‘Ž πœ™ ) c o s
  • E ( 0 , πœƒ , π‘Ž πœ™ ) c o t

Q5:

Convert the point ο€» √ 2 1 , βˆ’ √ 7 , 0  from Cartesian coordinates to cylindrical and spherical coordinates.

  • A ο€Ό 2 √ 7 , 1 1 πœ‹ 6 , 0  , ο€Ό 2 √ 7 , 1 1 πœ‹ 6 , πœ‹ 2 
  • B ο€» 7 √ 1 0 , βˆ’ πœ‹ 6 , 0  , ο€Ό 7 √ 1 0 , βˆ’ πœ‹ 6 , 2 πœ‹ 3 
  • C ο€» 2 √ 7 , πœ‹ 6 , 0  , ο€» 2 √ 7 , πœ‹ 6 , πœ‹ 2 
  • D ο€Ό 7 √ 1 0 , 1 1 πœ‹ 6 , 0  , ο€Ό 7 √ 1 0 , 1 1 πœ‹ 6 , πœ‹ 2 
  • E ο€Ό 2 √ 7 , 1 1 πœ‹ 6 , 0  , ο€Ό 2 √ 7 , 1 1 πœ‹ 6 , 2 πœ‹ 3 

Q6:

Write the given equation π‘₯ + 𝑦 + 𝑧 = 2 5 2 2 2 in cylindrical and spherical coordinates.

  • A 𝑧 + 𝜌 = 2 5 2 2 , π‘Ÿ = 5
  • B 𝑧 + 𝜌 = 2 5 2 , π‘Ÿ = 5
  • C 𝑧 βˆ’ 𝜌 = 2 5 2 2 , π‘Ÿ = 5
  • D 𝑧 βˆ’ 𝜌 = 2 5 2 , π‘Ÿ = 2 5
  • E 𝑧 + 𝜌 = 2 5 2 , π‘Ÿ = 2 5
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