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Calculus 3 : Calculus 3

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Example Questions

Example Question #283 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (62,-141,-67).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(79,-66.26^{\circ},-67)$

$(154.03,113.74^{\circ},-67)$

$(79,113.74^{\circ},-67)$

$(154.03,-66.26^{\circ},-67)$

Correct answer:

$(154.03,-66.26^{\circ},-67)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(62,-141,-67)\\&r=\sqrt{(62)^2+(-141)^2}=154.03\\&\theta=arctan(\frac{-141}{62})=-66.26^{\circ}\\&z=-67\end{align*}$

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Example Question #284 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-137,-121,97).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(182.78,41.45^{\circ},97)$

$(258,41.45^{\circ},97)$

$(182.78,-138.55^{\circ},97)$

$(258,-138.55^{\circ},97)$

Correct answer:

$(182.78,-138.55^{\circ},97)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-137,-121,97)\\&r=\sqrt{(-137)^2+(-121)^2}=182.78\\&\theta=arctan(\frac{-121}{-137})=-138.55^{\circ}\\&z=97\end{align*}$

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Example Question #201 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (59,-55,136).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(80.66,137.01^{\circ},136)$

$(4,137.01^{\circ},136)$

$(80.66,-42.99^{\circ},136)$

$(4,-42.99^{\circ},136)$

Correct answer:

$(80.66,-42.99^{\circ},136)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(59,-55,136)\\&r=\sqrt{(59)^2+(-55)^2}=80.66\\&\theta=arctan(\frac{-55}{59})=-42.99^{\circ}\\&z=136\end{align*}$

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Example Question #202 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-140,-18,-36).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(141.15,7.33^{\circ},-36)$

$(141.15,-172.67^{\circ},-36)$

$(158,-172.67^{\circ},-36)$

$(158,7.33^{\circ},-36)$

Correct answer:

$(141.15,-172.67^{\circ},-36)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-140,-18,-36)\\&r=\sqrt{(-140)^2+(-18)^2}=141.15\\&\theta=arctan(\frac{-18}{-140})=-172.67^{\circ}\\&z=-36\end{align*}$

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Example Question #203 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (80,89,-94).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(169,-131.95^{\circ},-94)$

$(119.67,48.05^{\circ},-94)$

$(119.67,-131.95^{\circ},-94)$

$(169,48.05^{\circ},-94)$

Correct answer:

$(119.67,48.05^{\circ},-94)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(80,89,-94)\\&r=\sqrt{(80)^2+(89)^2}=119.67\\&\theta=arctan(\frac{89}{80})=48.05^{\circ}\\&z=-94\end{align*}$

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Example Question #204 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-3,-16,44).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(16.28,-100.62^{\circ},44)$

$(16.28,79.38^{\circ},44)$

$(19,79.38^{\circ},44)$

$(19,-100.62^{\circ},44)$

Correct answer:

$(16.28,-100.62^{\circ},44)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-3,-16,44)\\&r=\sqrt{(-3)^2+(-16)^2}=16.28\\&\theta=arctan(\frac{-16}{-3})=-100.62^{\circ}\\&z=44\end{align*}$

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Example Question #291 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (63,77,-67).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(99.49,50.71^{\circ},-67)$

$(140,50.71^{\circ},-67)$

$(99.49,-129.29^{\circ},-67)$

$(140,-129.29^{\circ},-67)$

Correct answer:

$(99.49,50.71^{\circ},-67)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(63,77,-67)\\&r=\sqrt{(63)^2+(77)^2}=99.49\\&\theta=arctan(\frac{77}{63})=50.71^{\circ}\\&z=-67\end{align*}$

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Example Question #292 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (54,47,-102).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(101,-138.96^{\circ},-102)$

$(71.59,41.04^{\circ},-102)$

$(101,41.04^{\circ},-102)$

$(71.59,-138.96^{\circ},-102)$

Correct answer:

$(71.59,41.04^{\circ},-102)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(54,47,-102)\\&r=\sqrt{(54)^2+(47)^2}=71.59\\&\theta=arctan(\frac{47}{54})=41.04^{\circ}\\&z=-102\end{align*}$

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Example Question #293 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-48,26,-83).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(54.59,-28.44^{\circ},-83)$

$(22,151.56^{\circ},-83)$

$(54.59,151.56^{\circ},-83)$

$(22,-28.44^{\circ},-83)$

Correct answer:

$(54.59,151.56^{\circ},-83)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-48,26,-83)\\&r=\sqrt{(-48)^2+(26)^2}=54.59\\&\theta=arctan(\frac{26}{-48})=151.56^{\circ}\\&z=-83\end{align*}$

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Example Question #294 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (76,-74,2).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(106.08,-44.24^{\circ},2)$

$(2,135.76^{\circ},2)$

$(106.08,135.76^{\circ},2)$

$(2,-44.24^{\circ},2)$

Correct answer:

$(106.08,-44.24^{\circ},2)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(76,-74,2)\\&r=\sqrt{(76)^2+(-74)^2}=106.08\\&\theta=arctan(\frac{-74}{76})=-44.24^{\circ}\\&z=2\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #283 : 3 Dimensional Space

Example Question #284 : 3 Dimensional Space

Example Question #201 : Cylindrical Coordinates

Example Question #202 : Cylindrical Coordinates

Example Question #203 : Cylindrical Coordinates

Example Question #204 : Cylindrical Coordinates

Example Question #291 : 3 Dimensional Space

Example Question #292 : 3 Dimensional Space

Example Question #293 : 3 Dimensional Space

Example Question #294 : 3 Dimensional Space

All Calculus 3 Resources

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