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Calculus 3 : Calculus 3

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Example Questions

Example Question #221 : 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:

(80.43,67.49,60)

(67.49,80.43,60)

(-27.55,101.32,60)

(101.32,-27.55,60)

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 #140 : Cylindrical Coordinates

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

Possible Answers:

(-18.66,139.76,-111)

(139.76,-18.66,-111)

(59.59,127.79,-111)

(127.79,59.59,-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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Example Question #1891 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (95,122,-112).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(154.63,127.91^{\circ},-112)$

$(217,127.91^{\circ},-112)$

$(217,52.09^{\circ},-112)$

$(154.63,52.09^{\circ},-112)$

Correct answer:

$(154.63,52.09^{\circ},-112)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(95,122,-112)\\&r=\sqrt{(95)^2+(122)^2}=154.63\\&\theta=arctan(\frac{122}{95})=52.09^{\circ}\\&z=-112\end{align*}$

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Example Question #222 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (124,40,-121).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(130.29,17.88^{\circ},-121)$

$(130.29,162.12^{\circ},-121)$

$(164,162.12^{\circ},-121)$

$(164,17.88^{\circ},-121)$

Correct answer:

$(130.29,17.88^{\circ},-121)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(124,40,-121)\\&r=\sqrt{(124)^2+(40)^2}=130.29\\&\theta=arctan(\frac{40}{124})=17.88^{\circ}\\&z=-121\end{align*}$

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Example Question #1892 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-67,14,138).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(68.45,168.2^{\circ},138)$

$(53,168.2^{\circ},138)$

$(53,11.8^{\circ},138)$

$(68.45,11.8^{\circ},138)$

Correct answer:

$(68.45,168.2^{\circ},138)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-67,14,138)\\&r=\sqrt{(-67)^2+(14)^2}=68.45\\&\theta=arctan(\frac{14}{-67})=168.2^{\circ}\\&z=138\end{align*}$

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Example Question #144 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (140,-103,142).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(173.81,-36.34^{\circ},142)$

$(37,143.66^{\circ},142)$

$(173.81,143.66^{\circ},142)$

$(37,-36.34^{\circ},142)$

Correct answer:

$(173.81,-36.34^{\circ},142)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(140,-103,142)\\&r=\sqrt{(140)^2+(-103)^2}=173.81\\&\theta=arctan(\frac{-103}{140})=-36.34^{\circ}\\&z=142\end{align*}$

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Example Question #231 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (138,-4,90).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(138.06,178.34^{\circ},90)$

$(138.06,-1.66^{\circ},90)$

$(134,178.34^{\circ},90)$

$(134,-1.66^{\circ},90)$

Correct answer:

$(138.06,-1.66^{\circ},90)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(138,-4,90)\\&r=\sqrt{(138)^2+(-4)^2}=138.06\\&\theta=arctan(\frac{-4}{138})=-1.66^{\circ}\\&z=90\end{align*}$

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Example Question #141 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-108,-24,125).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(132,-167.47^{\circ},125)$

$(110.63,-167.47^{\circ},125)$

$(132,12.53^{\circ},125)$

$(110.63,12.53^{\circ},125)$

Correct answer:

$(110.63,-167.47^{\circ},125)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-108,-24,125)\\&r=\sqrt{(-108)^2+(-24)^2}=110.63\\&\theta=arctan(\frac{-24}{-108})=-167.47^{\circ}\\&z=125\end{align*}$

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Example Question #147 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (88,138,47).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(226,57.48^{\circ},47)$

$(163.67,122.52^{\circ},47)$

$(163.67,57.48^{\circ},47)$

$(226,122.52^{\circ},47)$

Correct answer:

$(163.67,57.48^{\circ},47)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(88,138,47)\\&r=\sqrt{(88)^2+(138)^2}=163.67\\&\theta=arctan(\frac{138}{88})=57.48^{\circ}\\&z=47\end{align*}$

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Example Question #148 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (54,78,73).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(132,55.3^{\circ},73)$

$(94.87,124.7^{\circ},73)$

$(94.87,55.3^{\circ},73)$

$(132,124.7^{\circ},73)$

Correct answer:

$(94.87,55.3^{\circ},73)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(54,78,73)\\&r=\sqrt{(54)^2+(78)^2}=94.87\\&\theta=arctan(\frac{78}{54})=55.3^{\circ}\\&z=73\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #221 : 3 Dimensional Space

Example Question #140 : Cylindrical Coordinates

Example Question #1891 : Calculus 3

Example Question #222 : 3 Dimensional Space

Example Question #1892 : Calculus 3

Example Question #144 : Cylindrical Coordinates

Example Question #231 : 3 Dimensional Space

Example Question #141 : Cylindrical Coordinates

Example Question #147 : Cylindrical Coordinates

Example Question #148 : Cylindrical Coordinates

All Calculus 3 Resources

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