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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #181 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (37,-54^{\circ},129).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-29.93,21.75,129)

(20.68,-30.68,129)

(-30.68,20.68,129)

(21.75,-29.93,129)

Correct answer:

(21.75,-29.93,129)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=37cos(-54^{\circ})=21.75\\&y=37sin(-54^{\circ})=-29.93\\&z=129\end{align*}$

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Example Question #182 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (30,42^{\circ},-75).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(22.29,20.07,-75)

(-12,-27.5,-75)

(20.07,22.29,-75)

(-27.5,-12,-75)

Correct answer:

(22.29,20.07,-75)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=30cos(42^{\circ})=22.29\\&y=30sin(42^{\circ})=20.07\\&z=-75\end{align*}$

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Example Question #101 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (71,119^{\circ},-45).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(65.92,-26.37,-45)

(-34.42,62.1,-45)

(62.1,-34.42,-45)

(-26.37,65.92,-45)

Correct answer:

(-34.42,62.1,-45)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=71cos(119^{\circ})=-34.42\\&y=71sin(119^{\circ})=62.1\\&z=-45\end{align*}$

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Example Question #102 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (88,151^{\circ},15).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(42.66,-76.97,15)

(86.18,17.79,15)

(-76.97,42.66,15)

(17.79,86.18,15)

Correct answer:

(-76.97,42.66,15)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=88cos(151^{\circ})=-76.97\\&y=88sin(151^{\circ})=42.66\\&z=15\end{align*}$

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Example Question #103 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (43,92^{\circ},77).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-26.94,-33.52,77)

(-1.5,42.97,77)

(42.97,-1.5,77)

(-33.52,-26.94,77)

Correct answer:

(-1.5,42.97,77)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=43cos(92^{\circ})=-1.5\\&y=43sin(92^{\circ})=42.97\\&z=77\end{align*}$

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Example Question #104 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (58,-153^{\circ},20).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-46.77,-34.3,20)

(-26.33,-51.68,20)

(-34.3,-46.77,20)

(-51.68,-26.33,20)

Correct answer:

(-51.68,-26.33,20)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=58cos(-153^{\circ})=-51.68\\&y=58sin(-153^{\circ})=-26.33\\&z=20\end{align*}$

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Example Question #191 : 3 Dimensional Space

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

Possible Answers:

(8.83,-1.72,9)

(8.03,4.07,9)

(4.07,8.03,9)

(-1.72,8.83,9)

Correct answer:

(-1.72,8.83,9)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=9cos(101^{\circ})=-1.72\\&y=9sin(101^{\circ})=8.83\\&z=9\end{align*}$

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Example Question #192 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (71,-59^{\circ},-147).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-45.21,-54.75,-147)

(-60.86,36.57,-147)

(-54.75,-45.21,-147)

(36.57,-60.86,-147)

Correct answer:

(36.57,-60.86,-147)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=71cos(-59^{\circ})=36.57\\&y=71sin(-59^{\circ})=-60.86\\&z=-147\end{align*}$

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Example Question #193 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (25,-68^{\circ},89).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(11,22.45,89)

(-23.18,9.37,89)

(22.45,11,89)

(9.37,-23.18,89)

Correct answer:

(9.37,-23.18,89)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=25cos(-68^{\circ})=9.37\\&y=25sin(-68^{\circ})=-23.18\\&z=89\end{align*}$

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Example Question #191 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in cylindrical coordinates, at } (80,37^{\circ},-101).\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(48.15,63.89,-101)

(-51.48,61.23,-101)

(63.89,48.15,-101)

(61.23,-51.48,-101)

Correct answer:

(63.89,48.15,-101)

Explanation:

$\begin{align*}&\text{If asked to convert cylindrical coordinates of the form} (r,\theta,z)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{ and }y\\& \text{to the radius, }r\text{, and the angle, }\theta.\text{ The relationships are as follows:}\\&x=rcos(\theta)\\&y=rsin(\theta)\\&\text{Finding }z\text{ is much simpler; it does not change between}\\& \text{Cartesian and cylindrical coordinates:}\\&z=z\\&\text{However, care should be taken when finding }x\text{ and }y\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=80cos(37^{\circ})=63.89\\&y=80sin(37^{\circ})=48.15\\&z=-101\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #181 : 3 Dimensional Space

Example Question #182 : 3 Dimensional Space

Example Question #101 : Cylindrical Coordinates

Example Question #102 : Cylindrical Coordinates

Example Question #103 : Cylindrical Coordinates

Example Question #104 : Cylindrical Coordinates

Example Question #191 : 3 Dimensional Space

Example Question #192 : 3 Dimensional Space

Example Question #193 : 3 Dimensional Space

Example Question #191 : 3 Dimensional Space

All Calculus 3 Resources

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