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Lesson: Cylindrical and Spherical Coordinates

Worksheet • 8 Questions

Q1:

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

Q2:

Convert the point ( βˆ’ 5 , 5 , 6 ) from Cartesian coordinates to cylindrical, ( π‘Ÿ , πœƒ , 𝑧 ) , and spherical, ( 𝜌 , πœƒ , πœ™ ) , coordinates. Round the value of πœ™ to two decimal places.

  • A ο€Ό 5 √ 2 , 3 πœ‹ 4 , 6  , ο€Ό √ 8 6 , 3 πœ‹ 4 , 4 9 . 6 8 
  • B ο€» 5 , βˆ’ πœ‹ 3 , 6  , ο€» 5 √ 2 , βˆ’ πœ‹ 3 , 3 0 . 0 0 
  • C ο€» √ 8 6 , πœ‹ 4 , 6  , ο€Ώ √ 8 6 , πœ‹ 4 , 5 √ 2 6 
  • D ο€Ό √ 8 6 , 3 πœ‹ 4 , 6  , ο€Ό √ 8 6 , 3 πœ‹ 4 , 5 4 . 7 3 
  • E ο€» 5 √ 2 , βˆ’ πœ‹ 4 , 6  , ο€» 5 √ 2 , βˆ’ πœ‹ 4 , 1 . 1 8 

Q3:

Convert the point ο€» 0 , √ 2 , 2  from Cartesian coordinates to cylindrical, ( π‘Ÿ , πœƒ , 𝑧 ) , and spherical, ( 𝜌 , πœƒ , πœ™ ) , coordinates. Round the value of πœ™ to two decimal places.

  • A ο€» √ 2 , πœ‹ 2 , 2  , ο€» √ 6 , πœ‹ 2 , 0 . 6 2 
  • B ο€Ό √ 2 , 3 πœ‹ 2 , 2  , ο€Ό √ 6 , 3 πœ‹ 2 , 0 . 6 8 
  • C ο€» 2 , πœ‹ 2 , 2  , ο€» 2 √ 2 , πœ‹ 2 , 0 . 6 8 
  • D ( 2 , πœ‹ , 2 ) , ο€» 2 √ 2 , πœ‹ , 0 . 6 2 
  • E ο€» √ 2 , πœ‹ , 2  , ο€» √ 6 , πœ‹ , 0 . 6 8 
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