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Calculus 3 : Calculus 3

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Example Questions

Example Question #178 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (84,131,-111).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(215,57.33^{\circ},-111)$

$(215,122.67^{\circ},-111)$

$(155.62,122.67^{\circ},-111)$

$(155.62,57.33^{\circ},-111)$

Correct answer:

$(155.62,57.33^{\circ},-111)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(84,131,-111)\\&r=\sqrt{(84)^2+(131)^2}=155.62\\&\theta=arctan(\frac{131}{84})=57.33^{\circ}\\&z=-111\end{align*}$

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Example Question #179 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (21,-9,-147).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(22.85,156.8^{\circ},-147)$

$(22.85,-23.2^{\circ},-147)$

$(12,-23.2^{\circ},-147)$

$(12,156.8^{\circ},-147)$

Correct answer:

$(22.85,-23.2^{\circ},-147)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(21,-9,-147)\\&r=\sqrt{(21)^2+(-9)^2}=22.85\\&\theta=arctan(\frac{-9}{21})=-23.2^{\circ}\\&z=-147\end{align*}$

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Example Question #1931 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1932 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1933 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1934 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1935 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1931 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1932 : Calculus 3

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

Possible Answers:

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

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

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

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

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 #1931 : Calculus 3

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

Possible Answers:

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

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

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

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

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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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #178 : Cylindrical Coordinates

Example Question #179 : Cylindrical Coordinates

Example Question #1931 : Calculus 3

Example Question #1932 : Calculus 3

Example Question #1933 : Calculus 3

Example Question #1934 : Calculus 3

Example Question #1935 : Calculus 3

Example Question #1931 : Calculus 3

Example Question #1932 : Calculus 3

Example Question #1931 : Calculus 3

All Calculus 3 Resources

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