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Calculus 3 : Cylindrical Coordinates

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Example Questions

Example Question #301 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (51,106^{\circ},-102).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-14.06,49.02,-102)

(49.02,-14.06,-102)

(-37.08,35.01,-102)

(35.01,-37.08,-102)

Correct answer:

(-14.06,49.02,-102)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=51cos(106^{\circ})=-14.06\\&y=51sin(106^{\circ})=49.02\\&z=-102\end{align*}$

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Example Question #221 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (47,-121^{\circ},9).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-2.29,-46.94,9)

(-24.21,-40.29,9)

(-40.29,-24.21,9)

(-46.94,-2.29,9)

Correct answer:

(-24.21,-40.29,9)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=47cos(-121^{\circ})=-24.21\\&y=47sin(-121^{\circ})=-40.29\\&z=9\end{align*}$

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Example Question #223 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (91,56^{\circ},-71).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(75.44,50.89,-71)

(77.64,-47.46,-71)

(50.89,75.44,-71)

(-47.46,77.64,-71)

Correct answer:

(50.89,75.44,-71)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=91cos(56^{\circ})=50.89\\&y=91sin(56^{\circ})=75.44\\&z=-71\end{align*}$

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Example Question #1971 : Calculus 3

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (104,-18^{\circ},75).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(78.1,68.67,75)

(68.67,78.1,75)

(-32.14,98.91,75)

(98.91,-32.14,75)

Correct answer:

(98.91,-32.14,75)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=104cos(-18^{\circ})=98.91\\&y=104sin(-18^{\circ})=-32.14\\&z=75\end{align*}$

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Example Question #222 : Cylindrical Coordinates

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

Possible Answers:

$(154.63,37.91^{\circ},-112)$

$(154.63,-142.09^{\circ},-112)$

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

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

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 #226 : Cylindrical Coordinates

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

Possible Answers:

$(130.29,72.12^{\circ},-121)$

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

$(130.29,-107.88^{\circ},-121)$

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

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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Example Question #223 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-67,14,138).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(68.45,101.8^{\circ},138)$

$(68.45,-11.8^{\circ},138)$

$(68.45,168.2^{\circ},138)$

$(68.45,-78.2^{\circ},138)$

Correct answer:

$(68.45,168.2^{\circ},138)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-67,14,138)\\&r=\sqrt{(-67)^2+(14)^2}=68.45\\&\theta=arctan(\frac{14}{-67})=168.2^{\circ}\\&z=138\end{align*}$

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Example Question #223 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (140,-103,142).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(173.81,-36.34^{\circ},142)$

$(173.81,-53.66^{\circ},142)$

$(173.81,143.66^{\circ},142)$

$(173.81,126.34^{\circ},142)$

Correct answer:

$(173.81,-36.34^{\circ},142)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(140,-103,142)\\&r=\sqrt{(140)^2+(-103)^2}=173.81\\&\theta=arctan(\frac{-103}{140})=-36.34^{\circ}\\&z=142\end{align*}$

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Example Question #311 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (138,-4,90).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(138.06,-88.34^{\circ},90)$

$(138.06,-1.66^{\circ},90)$

$(138.06,91.66^{\circ},90)$

$(138.06,178.34^{\circ},90)$

Correct answer:

$(138.06,-1.66^{\circ},90)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(138,-4,90)\\&r=\sqrt{(138)^2+(-4)^2}=138.06\\&\theta=arctan(\frac{-4}{138})=-1.66^{\circ}\\&z=90\end{align*}$

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Example Question #230 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-108,-24,125).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(110.63,-102.53^{\circ},125)$

$(110.63,77.47^{\circ},125)$

$(110.63,-167.47^{\circ},125)$

$(110.63,12.53^{\circ},125)$

Correct answer:

$(110.63,-167.47^{\circ},125)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-108,-24,125)\\&r=\sqrt{(-108)^2+(-24)^2}=110.63\\&\theta=arctan(\frac{-24}{-108})=-167.47^{\circ}\\&z=125\end{align*}$

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #301 : 3 Dimensional Space

Example Question #221 : Cylindrical Coordinates

Example Question #223 : Cylindrical Coordinates

Example Question #1971 : Calculus 3

Example Question #222 : Cylindrical Coordinates

Example Question #226 : Cylindrical Coordinates

Example Question #223 : Cylindrical Coordinates

Example Question #223 : Cylindrical Coordinates

Example Question #311 : 3 Dimensional Space

Example Question #230 : Cylindrical Coordinates

All Calculus 3 Resources

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