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Calculus 3 : Cylindrical Coordinates

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Example Questions

Example Question #181 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-49,-102,89).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(113.16,64.34^{\circ},89)$

$(151,64.34^{\circ},89)$

$(151,-115.66^{\circ},89)$

$(113.16,-115.66^{\circ},89)$

Correct answer:

$(113.16,-115.66^{\circ},89)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-49,-102,89)\\&r=\sqrt{(-49)^2+(-102)^2}=113.16\\&\theta=arctan(\frac{-102}{-49})=-115.66^{\circ}\\&z=89\end{align*}$

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Example Question #182 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-57,9,-101).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(48,8.97^{\circ},-101)$

$(57.71,8.97^{\circ},-101)$

$(57.71,171.03^{\circ},-101)$

$(48,171.03^{\circ},-101)$

Correct answer:

$(57.71,171.03^{\circ},-101)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-57,9,-101)\\&r=\sqrt{(-57)^2+(9)^2}=57.71\\&\theta=arctan(\frac{9}{-57})=171.03^{\circ}\\&z=-101\end{align*}$

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Example Question #183 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (31,-71,46).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(77.47,-66.41^{\circ},46)$

$(40,-66.41^{\circ},46)$

$(40,113.59^{\circ},46)$

$(77.47,113.59^{\circ},46)$

Correct answer:

$(77.47,-66.41^{\circ},46)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(31,-71,46)\\&r=\sqrt{(31)^2+(-71)^2}=77.47\\&\theta=arctan(\frac{-71}{31})=-66.41^{\circ}\\&z=46\end{align*}$

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Example Question #184 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (57,75,-15).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(94.2,52.77^{\circ},-15)$

$(94.2,127.23^{\circ},-15)$

$(132,127.23^{\circ},-15)$

$(132,52.77^{\circ},-15)$

Correct answer:

$(94.2,52.77^{\circ},-15)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(57,75,-15)\\&r=\sqrt{(57)^2+(75)^2}=94.2\\&\theta=arctan(\frac{75}{57})=52.77^{\circ}\\&z=-15\end{align*}$

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Example Question #271 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-125,-82,124).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(149.5,-146.74^{\circ},124)$

$(149.5,33.26^{\circ},124)$

$(207,33.26^{\circ},124)$

$(207,-146.74^{\circ},124)$

Correct answer:

$(149.5,-146.74^{\circ},124)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-125,-82,124)\\&r=\sqrt{(-125)^2+(-82)^2}=149.5\\&\theta=arctan(\frac{-82}{-125})=-146.74^{\circ}\\&z=124\end{align*}$

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Example Question #272 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-105,98,12).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(143.63,43.03^{\circ},12)$

$(7,43.03^{\circ},12)$

$(7,136.97^{\circ},12)$

$(143.63,136.97^{\circ},12)$

Correct answer:

$(143.63,136.97^{\circ},12)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-105,98,12)\\&r=\sqrt{(-105)^2+(98)^2}=143.63\\&\theta=arctan(\frac{98}{-105})=136.97^{\circ}\\&z=12\end{align*}$

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Example Question #273 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (149,-127,-17).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(22,-40.44^{\circ},-17)$

$(22,139.56^{\circ},-17)$

$(195.78,139.56^{\circ},-17)$

$(195.78,-40.44^{\circ},-17)$

Correct answer:

$(195.78,-40.44^{\circ},-17)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(149,-127,-17)\\&r=\sqrt{(149)^2+(-127)^2}=195.78\\&\theta=arctan(\frac{-127}{149})=-40.44^{\circ}\\&z=-17\end{align*}$

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Example Question #274 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-118,139,-149).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(182.33,-49.67^{\circ},-149)$

$(21,-49.67^{\circ},-149)$

$(182.33,130.33^{\circ},-149)$

$(21,130.33^{\circ},-149)$

Correct answer:

$(182.33,130.33^{\circ},-149)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-118,139,-149)\\&r=\sqrt{(-118)^2+(139)^2}=182.33\\&\theta=arctan(\frac{139}{-118})=130.33^{\circ}\\&z=-149\end{align*}$

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Example Question #275 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (95,122,-112).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(154.63,-127.91^{\circ},-112)$

$(217,-127.91^{\circ},-112)$

$(154.63,52.09^{\circ},-112)$

$(217,52.09^{\circ},-112)$

Correct answer:

$(154.63,52.09^{\circ},-112)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(95,122,-112)\\&r=\sqrt{(95)^2+(122)^2}=154.63\\&\theta=arctan(\frac{122}{95})=52.09^{\circ}\\&z=-112\end{align*}$

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Example Question #276 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (124,40,-121).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(164,17.88^{\circ},-121)$

$(164,-162.12^{\circ},-121)$

$(130.29,17.88^{\circ},-121)$

$(130.29,-162.12^{\circ},-121)$

Correct answer:

$(130.29,17.88^{\circ},-121)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(124,40,-121)\\&r=\sqrt{(124)^2+(40)^2}=130.29\\&\theta=arctan(\frac{40}{124})=17.88^{\circ}\\&z=-121\end{align*}$

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #181 : Cylindrical Coordinates

Example Question #182 : Cylindrical Coordinates

Example Question #183 : Cylindrical Coordinates

Example Question #184 : Cylindrical Coordinates

Example Question #271 : 3 Dimensional Space

Example Question #272 : 3 Dimensional Space

Example Question #273 : 3 Dimensional Space

Example Question #274 : 3 Dimensional Space

Example Question #275 : 3 Dimensional Space

Example Question #276 : 3 Dimensional Space

All Calculus 3 Resources

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