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.

  • A𝜌+9𝑧=36, 𝑟1+8𝜙=36cos
  • B𝜌+9𝑧=36, 𝑟1+8𝜙=36cos
  • C𝜌+9𝑧=36, 𝑟=36
  • D𝜌+9𝑧=36, 𝑟1+8𝜙=36sin
  • E𝜌+9𝑧=36, 𝑟=36

Q2:

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

  • A4,𝜋3,1, 17,𝜋3,0.57
  • B17,2𝜋3,1, 17,2𝜋3,1.82
  • C4,2𝜋3,1, 17,2𝜋3,0.57
  • D4,𝜋3,1, 17,𝜋3,1.82
  • E4,𝜋6,1, 17,𝜋6,1.82

Q3:

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

  • A𝜌=2𝜃sin, 𝑟𝜙=2𝜃sinsin
  • B𝜌=𝜃cos, 𝑟𝜙=𝜃sincos
  • C𝜌=2𝜃cos, 𝑟𝜙=2𝜃sincos
  • D𝜌=2𝜃sin, 𝑟𝜙=2𝜃sinsin
  • E𝜌=𝜃sin, 𝑟𝜙=𝜃sinsin

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,𝑎𝜙)cos
  • B(0,𝜃,𝑎𝜙)cot
  • C(𝑎,𝜃,𝑎𝜙)cot
  • D(𝑎,𝜃,𝑎𝜙)cos
  • E(𝑎,𝜙,𝑎𝜃)cot

Q5:

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

  • A27,11𝜋6,0, 27,11𝜋6,2𝜋3
  • B710,𝜋6,0, 710,𝜋6,2𝜋3
  • C710,11𝜋6,0, 710,11𝜋6,𝜋2
  • D27,11𝜋6,0, 27,11𝜋6,𝜋2
  • E27,𝜋6,0, 27,𝜋6,𝜋2

Q6:

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

  • A𝑧𝜌=25, 𝑟=25
  • B𝑧+𝜌=25, 𝑟=5
  • C𝑧+𝜌=25, 𝑟=5
  • D𝑧+𝜌=25, 𝑟=25
  • E𝑧𝜌=25, 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=𝑎𝑟:
  • B(𝑥,𝑦,𝑧)𝑥+𝑦=𝑎𝑧:
  • C(𝑟,𝜃,𝑧)𝑧=𝑎𝑟:
  • 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.

  • A2,𝜋2,2, 22,𝜋2,0.68
  • B2,𝜋,2, 6,𝜋,0.68
  • C2,3𝜋2,2, 6,3𝜋2,0.68
  • D2,𝜋2,2, 6,𝜋2,0.62
  • E(2,𝜋,2), 22,𝜋,0.62

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.

  • A52,𝜋4,6, 52,𝜋4,1.18
  • B86,3𝜋4,6, 86,3𝜋4,54.73
  • C52,3𝜋4,6, 86,3𝜋4,49.68
  • D5,𝜋3,6, 52,𝜋3,30.00
  • E86,𝜋4,6, 86,𝜋4,526

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