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Calculus 3 : Cylindrical Coordinates

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Example Questions

Example Question #157 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (20,14,25) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(34,34.99^{\circ},25)$

$(24.41,34.99^{\circ},25)$

$(24.41,145.01^{\circ},25)$

$(34,-145.01^{\circ},25)$

$(34,145.01^{\circ},25)$

Correct answer:

$(24.41,34.99^{\circ},25)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative values by convention create a negative $\theta$ , while negative values lead to $|\theta|>90^{\circ};\frac{\pi}{2}$

For our coordinates (20,14,25)

$\begin{align*}&r=\sqrt{(20)^2+(14)^2}=24.41\\&\theta=arctan(\frac{14}{20})=34.99^{\circ}\\&z=25\end{align*}$

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Example Question #158 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-9,12,15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(15,126.87^{\circ},15)$

$(15,-53.13^{\circ},15)$

$(3,126.87^{\circ},15)$

$(15,53.13^{\circ},15)$

$(3,-53.13^{\circ},15)$

Correct answer:

$(15,126.87^{\circ},15)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-9,12,15)

$\begin{align*}&r=\sqrt{(-9)^2+(12)^2}=15\\&\theta=arctan(\frac{12}{-9})=126.87^{\circ}\\&z=15\end{align*}$

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Example Question #159 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (36,-160,18) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(124,-77.32^{\circ},18)$

$(124,102.68^{\circ},18)$

$(164,102.68^{\circ},18)$

$(124,77.32^{\circ},18)$

$(164,-77.32^{\circ},18)$

Correct answer:

$(164,-77.32^{\circ},18)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (36,-160,18)

$\begin{align*}&r=\sqrt{(36)^2+(-160)^2}=164\\&\theta=arctan(\frac{-160}{36})=-77.32^{\circ}\\&z=18\end{align*}$

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Example Question #160 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-1.5,-2,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(2.5,126.87^{\circ},10)$

$(3.5,-126.87^{\circ},10)$

$(2.5,53.13^{\circ},10)$

$(3.5,53.13^{\circ},10)$

$(2.5,-126.87^{\circ},10)$

Correct answer:

$(2.5,-126.87^{\circ},10)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-1.5,-2,10)

$\begin{align*}&r=\sqrt{(-1.5)^2+(-2)^2}=2.5\\&\theta=arctan(\frac{-2}{-1.5})=-126.87^{\circ}\\&z=10\end{align*}$

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Example Question #161 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-50,120,100) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(70,67.38^{\circ},100)$

$(130,112.62^{\circ},100)$

$(70,-67.38^{\circ},100)$

$(130,-67.38^{\circ},100)$

$(70,112.62^{\circ},100)$

Correct answer:

$(130,112.62^{\circ},100)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-50,120,100)

$\begin{align*}&r=\sqrt{(-50)^2+(120)^2}=130\\&\theta=arctan(\frac{120}{-50})=112.62^{\circ}\\&z=100\end{align*}$

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Example Question #162 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (5,-25,-6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(25.5,78.69^{\circ},-6)$

$(25.5,-78.69^{\circ},-6)$

$(20,-78.69^{\circ},-6)$

$(25.5,101.31^{\circ},-6)$

$(20,101.31^{\circ},-6)$

Correct answer:

$(25.5,-78.69^{\circ},-6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (5,-25,-6)

$\begin{align*}&r=\sqrt{(5)^2+(-25)^2}=25.5\\&\theta=arctan(\frac{-25}{5})=-78.69^{\circ}\\&z=-6\end{align*}$

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Example Question #163 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (1,-1.707,6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(0.71,120.36^{\circ},6)$

$(1.98,120.36^{\circ},6)$

$(0.71,-120.36^{\circ},6)$

$(1.98,-59.64^{\circ},6)$

$(0.71,-59.64^{\circ},6)$

Correct answer:

$(1.98,-59.64^{\circ},6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (1,-1.707,6)

$\begin{align*}&r=\sqrt{(1)^2+(-1.71)^2}=1.98\\&\theta=arctan(\frac{-1.71}{1})=-59.64^{\circ}\\&z=6\end{align*}$

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Example Question #164 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (7,21,22) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(28,108.43^{\circ},22)$

$(22.14,-71.57^{\circ},22)$

$(22.14,71.57^{\circ},22)$

$(22.14,108.43^{\circ},22)$

$(28,71.57^{\circ},22)$

Correct answer:

$(22.14,71.57^{\circ},22)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (7,21,22)

$\begin{align*}&r=\sqrt{(7)^2+(21)^2}=22.14\\&\theta=arctan(\frac{21}{7})=71.57^{\circ}\\&z=22\end{align*}$

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Example Question #165 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (3,24,5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(24.19,82.87^{\circ},5)$

$(24.19,97.13^{\circ},5)$

$(27,-82.87^{\circ},5)$

$(27,97.13^{\circ},5)$

$(27,82.87^{\circ},5)$

Correct answer:

$(24.19,82.87^{\circ},5)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (3,24,5)

$\begin{align*}&r=\sqrt{(3)^2+(24)^2}=24.19\\&\theta=arctan(\frac{24}{3})=82.87^{\circ}\\&z=5\end{align*}$

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Example Question #166 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-2,-20,-22) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(22,84.29^{\circ},-22)$

$(20.1,95.71^{\circ},-22)$

$(20.1,84.29^{\circ},-22)$

$(20.1,-95.71^{\circ},-22)$

$(20.1,-84.29^{\circ},-22)$

Correct answer:

$(20.1,-95.71^{\circ},-22)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-2,-20,-22)

$\begin{align*}&r=\sqrt{(-2)^2+(-20)^2}=20.1\\&\theta=arctan(\frac{-20}{-2})=-95.71^{\circ}\\&z=-22\end{align*}$

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #157 : 3 Dimensional Space

Example Question #158 : 3 Dimensional Space

Example Question #159 : 3 Dimensional Space

Example Question #160 : 3 Dimensional Space

Example Question #161 : 3 Dimensional Space

Example Question #162 : 3 Dimensional Space

Example Question #163 : 3 Dimensional Space

Example Question #164 : 3 Dimensional Space

Example Question #165 : 3 Dimensional Space

Example Question #166 : 3 Dimensional Space

All Calculus 3 Resources

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