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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #155 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (74,53^{\circ},-16).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(59.1,44.53,-16)

(29.3,-67.95,-16)

(44.53,59.1,-16)

(-67.95,29.3,-16)

Correct answer:

(44.53,59.1,-16)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=74cos(53^{\circ})=44.53\\&y=74sin(53^{\circ})=59.1\\&z=-16\end{align*}$

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Example Question #156 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (107,-81^{\circ},77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(67.4,83.11,77)

(83.11,67.4,77)

(16.74,-105.68,77)

(-105.68,16.74,77)

Correct answer:

(16.74,-105.68,77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=107cos(-81^{\circ})=16.74\\&y=107sin(-81^{\circ})=-105.68\\&z=77\end{align*}$

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Example Question #157 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (102,-122^{\circ},47).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-86.5,-54.05,47)

(-50.87,-88.41,47)

(-54.05,-86.5,47)

(-88.41,-50.87,47)

Correct answer:

(-54.05,-86.5,47)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=102cos(-122^{\circ})=-54.05\\&y=102sin(-122^{\circ})=-86.5\\&z=47\end{align*}$

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Example Question #151 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (18,166^{\circ},0).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(4.35,-17.47,0)

(8.7,-15.76,0)

(-15.76,8.7,0)

(-17.47,4.35,0)

Correct answer:

(-17.47,4.35,0)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=18cos(166^{\circ})=-17.47\\&y=18sin(166^{\circ})=4.35\\&z=0\end{align*}$

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Example Question #155 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (52,-100^{\circ},26).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-51.21,-9.03,26)

(-9.03,-51.21,26)

(44.84,26.33,26)

(26.33,44.84,26)

Correct answer:

(-9.03,-51.21,26)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=52cos(-100^{\circ})=-9.03\\&y=52sin(-100^{\circ})=-51.21\\&z=26\end{align*}$

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Example Question #160 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (113,2^{\circ},-74).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-47.02,102.75,-74)

(112.93,3.94,-74)

(102.75,-47.02,-74)

(3.94,112.93,-74)

Correct answer:

(112.93,3.94,-74)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=113cos(2^{\circ})=112.93\\&y=113sin(2^{\circ})=3.94\\&z=-74\end{align*}$

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Example Question #161 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (83,-127^{\circ},-109).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-49.95,-66.29,-109)

(-80.73,19.29,-109)

(19.29,-80.73,-109)

(-66.29,-49.95,-109)

Correct answer:

(-49.95,-66.29,-109)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=83cos(-127^{\circ})=-49.95\\&y=83sin(-127^{\circ})=-66.29\\&z=-109\end{align*}$

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Example Question #162 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (39,-89^{\circ},103).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-38.99,0.68,103)

(-33.54,19.9,103)

(19.9,-33.54,103)

(0.68,-38.99,103)

Correct answer:

(0.68,-38.99,103)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=39cos(-89^{\circ})=0.68\\&y=39sin(-89^{\circ})=-38.99\\&z=103\end{align*}$

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Example Question #163 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (123,155^{\circ},-77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-107.42,-59.92,-77)

(-59.92,-107.42,-77)

(-111.48,51.98,-77)

(51.98,-111.48,-77)

Correct answer:

(-111.48,51.98,-77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=123cos(155^{\circ})=-111.48\\&y=123sin(155^{\circ})=51.98\\&z=-77\end{align*}$

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Example Question #164 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (53,-90^{\circ},-91).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-53,0,-91)

(0,-53,-91)

(-23.75,-47.38,-91)

(-47.38,-23.75,-91)

Correct answer:

(0,-53,-91)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=53cos(-90^{\circ})=0\\&y=53sin(-90^{\circ})=-53\\&z=-91\end{align*}$

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Calculus 3 : 3-Dimensional Space

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #155 : Cylindrical Coordinates

Example Question #156 : Cylindrical Coordinates

Example Question #157 : Cylindrical Coordinates

Example Question #151 : Cylindrical Coordinates

Example Question #155 : Cylindrical Coordinates

Example Question #160 : Cylindrical Coordinates

Example Question #161 : Cylindrical Coordinates

Example Question #162 : Cylindrical Coordinates

Example Question #163 : Cylindrical Coordinates

Example Question #164 : Cylindrical Coordinates

All Calculus 3 Resources

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