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Calculus 3 : Cylindrical Coordinates

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Example Questions

Example Question #217 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-75,-63,35).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(97.95,-139.97^{\circ},35)$

$(138,-139.97^{\circ},35)$

$(138,40.03^{\circ},35)$

$(97.95,40.03^{\circ},35)$

Correct answer:

$(97.95,-139.97^{\circ},35)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-75,-63,35)\\&r=\sqrt{(-75)^2+(-63)^2}=97.95\\&\theta=arctan(\frac{-63}{-75})=-139.97^{\circ}\\&z=35\end{align*}$

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Example Question #131 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-71,98,145).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(27,54.08^{\circ},145)$

$(27,125.92^{\circ},145)$

$(121.02,125.92^{\circ},145)$

$(121.02,54.08^{\circ},145)$

Correct answer:

$(121.02,125.92^{\circ},145)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-71,98,145)\\&r=\sqrt{(-71)^2+(98)^2}=121.02\\&\theta=arctan(\frac{98}{-71})=125.92^{\circ}\\&z=145\end{align*}$

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Example Question #219 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (69,-47,25).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(83.49,145.74^{\circ},25)$

$(83.49,-34.26^{\circ},25)$

$(22,-34.26^{\circ},25)$

$(22,145.74^{\circ},25)$

Correct answer:

$(83.49,-34.26^{\circ},25)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(69,-47,25)\\&r=\sqrt{(69)^2+(-47)^2}=83.49\\&\theta=arctan(\frac{-47}{69})=-34.26^{\circ}\\&z=25\end{align*}$

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Example Question #220 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (17,137^{\circ},122).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-12.43,11.59,122)

(-16.02,5.68,122)

(11.59,-12.43,122)

(5.68,-16.02,122)

Correct answer:

(-12.43,11.59,122)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=17cos(137^{\circ})=-12.43\\&y=17sin(137^{\circ})=11.59\\&z=122\end{align*}$

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Example Question #221 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (123,34^{\circ},-72).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(68.78,101.97,-72)

(101.97,68.78,-72)

(65.08,-104.37,-72)

(-104.37,65.08,-72)

Correct answer:

(101.97,68.78,-72)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=123cos(34^{\circ})=101.97\\&y=123sin(34^{\circ})=68.78\\&z=-72\end{align*}$

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Example Question #222 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (4,-68^{\circ},-22).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(3.59,1.76,-22)

(-3.71,1.5,-22)

(1.76,3.59,-22)

(1.5,-3.71,-22)

Correct answer:

(1.5,-3.71,-22)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=4cos(-68^{\circ})=1.5\\&y=4sin(-68^{\circ})=-3.71\\&z=-22\end{align*}$

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Example Question #223 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (25,-28^{\circ},-97).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-24.07,-6.77,-97)

(22.07,-11.74,-97)

(-6.77,-24.07,-97)

(-11.74,22.07,-97)

Correct answer:

(22.07,-11.74,-97)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=25cos(-28^{\circ})=22.07\\&y=25sin(-28^{\circ})=-11.74\\&z=-97\end{align*}$

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Example Question #224 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (15,-10^{\circ},30).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-2.6,14.77,30)

(-12.59,8.16,30)

(14.77,-2.6,30)

(8.16,-12.59,30)

Correct answer:

(14.77,-2.6,30)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=15cos(-10^{\circ})=14.77\\&y=15sin(-10^{\circ})=-2.6\\&z=30\end{align*}$

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Example Question #225 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (105,50^{\circ},60).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(101.32,-27.55,60)

(67.49,80.43,60)

(80.43,67.49,60)

(-27.55,101.32,60)

Correct answer:

(67.49,80.43,60)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=105cos(50^{\circ})=67.49\\&y=105sin(50^{\circ})=80.43\\&z=60\end{align*}$

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Example Question #226 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (141,25^{\circ},-111).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-18.66,139.76,-111)

(127.79,59.59,-111)

(139.76,-18.66,-111)

(59.59,127.79,-111)

Correct answer:

(127.79,59.59,-111)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=141cos(25^{\circ})=127.79\\&y=141sin(25^{\circ})=59.59\\&z=-111\end{align*}$

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #217 : 3 Dimensional Space

Example Question #131 : Cylindrical Coordinates

Example Question #219 : 3 Dimensional Space

Example Question #220 : 3 Dimensional Space

Example Question #221 : 3 Dimensional Space

Example Question #222 : 3 Dimensional Space

Example Question #223 : 3 Dimensional Space

Example Question #224 : 3 Dimensional Space

Example Question #225 : 3 Dimensional Space

Example Question #226 : 3 Dimensional Space

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