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Calculus 3 : Calculus 3

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Example Questions

Example Question #101 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (40,68^{\circ},46).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(37.09,14.98,46)

(-35.92,17.61,46)

(14.98,37.09,46)

(17.61,-35.92,46)

Correct answer:

(14.98,37.09,46)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=40cos(68^{\circ})=14.98\\&y=40sin(68^{\circ})=37.09\\&z=46\end{align*}$

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Example Question #193 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (113,-150^{\circ},-15).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-56.5,-97.86,-15)

(80.78,79.02,-15)

(-97.86,-56.5,-15)

(79.02,80.78,-15)

Correct answer:

(-97.86,-56.5,-15)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=113cos(-150^{\circ})=-97.86\\&y=113sin(-150^{\circ})=-56.5\\&z=-15\end{align*}$

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Example Question #111 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (35,-125^{\circ},124).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(27.57,21.56,124)

(-28.67,-20.08,124)

(-20.08,-28.67,124)

(21.56,27.57,124)

Correct answer:

(-20.08,-28.67,124)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=35cos(-125^{\circ})=-20.08\\&y=35sin(-125^{\circ})=-28.67\\&z=124\end{align*}$

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Example Question #111 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (124,179^{\circ},12).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-123.69,8.77,12)

(2.16,-123.98,12)

(8.77,-123.69,12)

(-123.98,2.16,12)

Correct answer:

(-123.98,2.16,12)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=124cos(179^{\circ})=-123.98\\&y=124sin(179^{\circ})=2.16\\&z=12\end{align*}$

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Example Question #111 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (12,-142^{\circ},-17).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-9.71,7.05,-17)

(-9.46,-7.39,-17)

(-7.39,-9.46,-17)

(7.05,-9.71,-17)

Correct answer:

(-9.46,-7.39,-17)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=12cos(-142^{\circ})=-9.46\\&y=12sin(-142^{\circ})=-7.39\\&z=-17\end{align*}$

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Example Question #192 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (145,99^{\circ},-149).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-22.68,143.21,-149)

(-144.88,5.77,-149)

(5.77,-144.88,-149)

(143.21,-22.68,-149)

Correct answer:

(-22.68,143.21,-149)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=145cos(99^{\circ})=-22.68\\&y=145sin(99^{\circ})=143.21\\&z=-149\end{align*}$

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Example Question #201 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (60,108^{\circ},-72).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(22.53,55.61,-72)

(55.61,22.53,-72)

(57.06,-18.54,-72)

(-18.54,57.06,-72)

Correct answer:

(-18.54,57.06,-72)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=60cos(108^{\circ})=-18.54\\&y=60sin(108^{\circ})=57.06\\&z=-72\end{align*}$

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Example Question #202 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (111,24,15).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(135,12.2^{\circ},15)$

$(135,167.8^{\circ},15)$

$(113.56,167.8^{\circ},15)$

$(113.56,12.2^{\circ},15)$

Correct answer:

$(113.56,12.2^{\circ},15)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms are the most straightforward }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{However, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\text{For our coordinates: }(111,24,15)\\&r=\sqrt{(111)^2+(24)^2}=113.56\\&\theta=arctan(\frac{24}{111})=12.2^{\circ}\\&z=15\end{align*}$

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Example Question #113 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-107,106,37).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(150.62,135.27^{\circ},37)$

$(150.62,44.73^{\circ},37)$

$(1,135.27^{\circ},37)$

$(1,44.73^{\circ},37)$

Correct answer:

$(150.62,135.27^{\circ},37)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-107,106,37)\\&r=\sqrt{(-107)^2+(106)^2}=150.62\\&\theta=arctan(\frac{106}{-107})=135.27^{\circ}\\&z=37\end{align*}$

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Example Question #114 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-45,4,-30).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(41,5.08^{\circ},-30)$

$(45.18,174.92^{\circ},-30)$

$(41,174.92^{\circ},-30)$

$(45.18,5.08^{\circ},-30)$

Correct answer:

$(45.18,174.92^{\circ},-30)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-45,4,-30)\\&r=\sqrt{(-45)^2+(4)^2}=45.18\\&\theta=arctan(\frac{4}{-45})=174.92^{\circ}\\&z=-30\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #101 : Cylindrical Coordinates

Example Question #193 : 3 Dimensional Space

Example Question #111 : Cylindrical Coordinates

Example Question #111 : Cylindrical Coordinates

Example Question #111 : Cylindrical Coordinates

Example Question #192 : 3 Dimensional Space

Example Question #201 : 3 Dimensional Space

Example Question #202 : 3 Dimensional Space

Example Question #113 : Cylindrical Coordinates

Example Question #114 : Cylindrical Coordinates

All Calculus 3 Resources

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