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Calculus 3 : Calculus 3

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

Example Questions

Example Question #9 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (6,8,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(10,-126.87^{\circ},10)$

$(10,53.13^{\circ},10)$

$(10,53.13^{\circ},20)$

$(10,-71.19^{\circ},10)$

$(10,108.81^{\circ},10)$

Correct answer:

$(10,53.13^{\circ},10)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative values by convention create a negative $\theta$ , while negative values lead to $|\theta|>90^{\circ};\frac{\pi}{2}$

For our coordinates (6,8,10)

$r=\sqrt{x^2+y^2}\rightarrow10$

$\theta=arctan(\frac{y}{x})\rightarrow 53.13^{\circ}$ (Bearing in mind sign convention)

z=10

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Example Question #10 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-5,-5,-5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(7.07,-135^{\circ},-5)$

$(7.07,70^{\circ},-5)$

$(7.07,45^{\circ},-5)$

$(7.07,-120^{\circ},-5)$

$(7.07,110^{\circ},-5)$

Correct answer:

$(7.07,-135^{\circ},-5)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-5,-5,-5)

$r=\sqrt{x^2+y^2}\rightarrow7.07$

$\theta=arctan(\frac{y}{x})\rightarrow -135^{\circ}$ (Bearing in mind sign convention)

z=-5

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Example Question #92 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at $(4,-4\sqrt{3},14)$ . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8,135^{\circ},14)$

$(8,-45^{\circ},14)$

$(8,-60^{\circ},14)$

$(8,120^{\circ},14)$

$(8,45^{\circ},14)$

Correct answer:

$(8,-60^{\circ},14)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates $(4,-4\sqrt{3},14)$

$r=\sqrt{x^2+y^2}\rightarrow 8$

$\theta=arctan(\frac{y}{x})\rightarrow -60^{\circ}$ (Bearing in mind sign convention)

z=14

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Example Question #93 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (9,12,6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(15,53.13^{\circ},6)$

$(21,126.87^{\circ},6)$

$(15,126.87^{\circ},6)$

$(21,53.13^{\circ},6)$

$(21,-126.87^{\circ},6)$

Correct answer:

$(15,53.13^{\circ},6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (9,12,6)

$r=\sqrt{x^2+y^2}\rightarrow 15$

$\theta=arctan(\frac{y}{x})\rightarrow 53.13^{\circ}$ (Bearing in mind sign convention)

z=6

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Example Question #94 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (10,13,16) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(16.4,52.43^{\circ},16)$

$(16.4,-127.57^{\circ},16)$

$(16.4,127.57^{\circ},16)$

$(23,-52.43^{\circ},16)$

$(23,127.57^{\circ},16)$

Correct answer:

$(16.4,52.43^{\circ},16)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (10,13,16)

$r=\sqrt{x^2+y^2}\rightarrow16.4$

$\theta=arctan(\frac{y}{x})\rightarrow 52.43^{\circ}$ (Bearing in mind sign convention)

z=16

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Example Question #95 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (17,49,62) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(51.9,70.87^{\circ},62)$

$(66,70.87^{\circ},62)$

$(66,-70.87^{\circ},62)$

$(51.9,-109.13^{\circ},62)$

$(51.9,109.13^{\circ},62)$

Correct answer:

$(51.9,70.87^{\circ},62)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (17,49,62)

$r=\sqrt{x^2+y^2}\rightarrow 51.9$

$\theta=arctan(\frac{y}{x})\rightarrow 70.87^{\circ}$ (Bearing in mind sign convention)

z=62

Report an Error

Example Question #101 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-8,5,0) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(11.23,-32.01^{\circ},0)$

$(9.43,-32.01^{\circ},0)$

$(9.43,32.01^{\circ},0)$

$(11.23,32.01^{\circ},0)$

$(9.43,147.99^{\circ},0)$

Correct answer:

$(9.43,147.99^{\circ},0)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-8,5,0)

$r=\sqrt{x^2+y^2}\rightarrow9.43$

$\theta=arctan(\frac{y}{x})\rightarrow 147.99^{\circ}$ (Bearing in mind sign convention)

z=0

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Example Question #102 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-6,-2,8) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(-8,18.44^{\circ},8)$

$(-8,-161.57^{\circ},8)$

$(6.32,-161.57^{\circ},8)$

$(-8,-18.44^{\circ},8)$

$(6.32,18.44^{\circ},8)$

Correct answer:

$(6.32,-161.57^{\circ},8)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-6,-2,8)

$r=\sqrt{x^2+y^2}\rightarrow6.32$

$\theta=arctan(\frac{y}{x})\rightarrow -161.57^{\circ}$ (Bearing in mind sign convention)

z=8

Report an Error

Example Question #103 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (4,7,3) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.06,-60.26^{\circ},3)$

$(8.06,119.74^{\circ},3)$

$(8.06,60.26^{\circ},3)$

$(8.06,81.32^{\circ},3)$

$(8.06,-119.74^{\circ},3)$

Correct answer:

$(8.06,60.26^{\circ},3)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (4,7,3)

$r=\sqrt{x^2+y^2}\rightarrow8.06$

$\theta=arctan(\frac{y}{x})\rightarrow 60.26^{\circ}$ (Bearing in mind sign convention)

z=3

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Example Question #104 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-8,4,-2) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.94,-54.83^{\circ},-2)$

$(8.94,-125.17^{\circ},-2)$

$(8.94,125.17^{\circ},-2)$

$(8.94,54.83^{\circ},-2)$

$(8.94,153.43^{\circ},-2)$

Correct answer:

$(8.94,153.43^{\circ},-2)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-8,4,-2)

$r=\sqrt{x^2+y^2}\rightarrow8.94$

$\theta=arctan(\frac{y}{x})\rightarrow 153.43^{\circ}$ (Bearing in mind sign convention)

z=-2

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #9 : Cylindrical Coordinates

Example Question #10 : Cylindrical Coordinates

Example Question #92 : 3 Dimensional Space

Example Question #93 : 3 Dimensional Space

Example Question #94 : 3 Dimensional Space

Example Question #95 : 3 Dimensional Space

Example Question #101 : 3 Dimensional Space

Example Question #102 : 3 Dimensional Space

Example Question #103 : 3 Dimensional Space

Example Question #104 : 3 Dimensional Space

All Calculus 3 Resources

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