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Calculus 3 : Calculus 3

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Example Questions

Example Question #505 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (37,-146^{\circ},73^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-2.1,24.95,-27.24)

(-8.97,-6.05,35.38)

(-2.29,27.14,-25.04)

(-29.33,-19.79,10.82)

Correct answer:

(-29.33,-19.79,10.82)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=37sin(73^{\circ})cos(-146^{\circ})=-29.33\\&y=37sin(73^{\circ})sin(-146^{\circ})=-19.79\\&z=37cos(73^{\circ})10.82\end{align*}$

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Example Question #506 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (20,165^{\circ},169^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-1.06,15.93,-12.04)

(-3.69,0.99,-19.63)

(0.8,-12.01,15.97)

(18.96,-5.08,3.82)

Correct answer:

(-3.69,0.99,-19.63)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=20sin(169^{\circ})cos(165^{\circ})=-3.69\\&y=20sin(169^{\circ})sin(165^{\circ})=0.99\\&z=20cos(169^{\circ})-19.63\end{align*}$

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Example Question #507 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (87,-96^{\circ},11^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-1.74,-16.51,85.4)

(15.7,85.57,0.39)

(-8.93,-84.93,16.6)

(-0.07,-0.38,-87)

Correct answer:

(-1.74,-16.51,85.4)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=87sin(11^{\circ})cos(-96^{\circ})=-1.74\\&y=87sin(11^{\circ})sin(-96^{\circ})=-16.51\\&z=87cos(11^{\circ})85.4\end{align*}$

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Example Question #508 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (53,-175^{\circ},147^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(-25.16,-33.68,32.28)

(44.28,3.87,28.87)

(19.32,25.86,-42.04)

(-28.76,-2.52,-44.45)

Correct answer:

(-28.76,-2.52,-44.45)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=53sin(147^{\circ})cos(-175^{\circ})=-28.76\\&y=53sin(147^{\circ})sin(-175^{\circ})=-2.52\\&z=53cos(147^{\circ})-44.45\end{align*}$

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Example Question #161 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (7,54^{\circ},31^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(2.35,1.58,6.4)

(2.12,2.92,6)

(-5.31,-3.58,-2.83)

(3.53,4.85,3.61)

Correct answer:

(2.12,2.92,6)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=7sin(31^{\circ})cos(54^{\circ})=2.12\\&y=7sin(31^{\circ})sin(54^{\circ})=2.92\\&z=7cos(31^{\circ})6\end{align*}$

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Example Question #509 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (110,-18^{\circ},116^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(17.19,19.55,-106.88)

(94.03,-30.55,-48.22)

(-70.57,-80.26,26.03)

(-45.86,14.9,98.87)

Correct answer:

(94.03,-30.55,-48.22)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=110sin(116^{\circ})cos(-18^{\circ})=94.03\\&y=110sin(116^{\circ})sin(-18^{\circ})=-30.55\\&z=110cos(116^{\circ})-48.22\end{align*}$

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Example Question #166 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (83,88^{\circ},54^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(1.7,48.76,67.15)

(-68.79,-2.44,-46.38)

(2.34,67.11,48.79)

(-46.35,-1.64,-68.83)

Correct answer:

(2.34,67.11,48.79)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=83sin(54^{\circ})cos(88^{\circ})=2.34\\&y=83sin(54^{\circ})sin(88^{\circ})=67.11\\&z=83cos(54^{\circ})48.79\end{align*}$

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Example Question #167 : Spherical Coordinates

$\begin{align*}&\text{A point in space is located, in spherical coordinates, at } (29,-114^{\circ},123^{\circ})\text{ of form: } (\rho,\theta,\phi)\\&\text{Where }\phi\text{ is the angle between the vector, }\rho\text{, to the point and the }z\text{ axis}.\\&\text{What is the position of this point in Cartesian coordinates?}\end{align*}$

Possible Answers:

(6.42,14.43,24.32)

(-15.95,20.21,-13.34)

(-8.26,10.47,-25.75)

(-9.89,-22.22,-15.79)

Correct answer:

(-9.89,-22.22,-15.79)

Explanation:

$\begin{align*}&\text{When converting spherical coordinates of the form} (\rho,\theta,\phi)\\&\text{to Cartesian coordinates of the form }(x,y,z)\text{, it is necessary to relate } x\text{, }y\text{, and }z\\& \text{to the vector, }\rho\text{, and the angles }\theta.\text{ and }\phi.\text{ The relationships are as follows:}\\&x=\rho sin(\phi)cos(\theta)\\&y=\rho sin(\phi)sin(\theta)\\&z=\rho cos(\phi)\\&\text{You may notice that }\rho sin(\phi)\text{ is the projection of }\rho\text{ on the }xy\text{-plane!}\\&\text{However, care should be taken when finding these terms}\text{; if using a calculator, it is}\\& \text{imperative that the correct units (degrees or radians) are specified}\\& \text{for the input!}\\&\text{For our coordinates}\\&x=29sin(123^{\circ})cos(-114^{\circ})=-9.89\\&y=29sin(123^{\circ})sin(-114^{\circ})=-22.22\\&z=29cos(123^{\circ})-15.79\end{align*}$

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Example Question #168 : Spherical Coordinates

Convert the following vector from Cartesian coordinates into spherical coordinates:

$(x^{-1}y, xz, 3y^2)$

Possible Answers:

$(\cot\phi, r^2 \sin\theta\cos^2 \theta, 3r^2\sin^2 \theta \sin^2 \phi )$

$(\tan \phi, r^2 \sin\theta\cos \phi \cos \theta, 3r^2\sin^2 \theta \sin^2 \phi )$

$(\tan \phi, r^2 \sin\theta\cos^2 \theta, 3r^2\sin^2 \theta \sin^2 \phi )$

$(\tan \phi, r^2 \sin\theta\cos^2 \phi , 3r^2\sin^2 \theta \sin^2 \phi )$

$(\cot\phi, r^2 \sin\theta\cos^2 \phi , 3r^2\sin^2 \theta \sin^2 \phi )$

Correct answer:

$(\tan \phi, r^2 \sin\theta\cos \phi \cos \theta, 3r^2\sin^2 \theta \sin^2 \phi )$

Explanation:

The conversion from Cartesian to spherical coordinates is as follows:

$x\rightarrow r\sin \theta \cos \phi$

$y\rightarrow r\sin \theta \sin\phi$

$z\rightarrow r\cos \phi$

The three components of the vector therefore become:

$\frac{y}{x}\rightarrow \frac{ r\sin \theta \sin\phi}{ r\sin \theta \cos\phi}=\tan \phi$

$xz\rightarrow( r \sin \theta \cos \phi )(r \cos \theta)=r^2 \sin\theta\cos \phi \cos \theta,$

$3y^2=3r^2 \sin^2 \theta \sin^2 \phi$

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Example Question #511 : 3 Dimensional Space

Find an equation of a sphere that has the line segment joining (0,4,2) and (6,0,2) as a diameter.

Possible Answers:

(x-2)^2+(y-3)^2+(z-2)^2=13

None of the Above.

(x-3)^2+(y-2)^2+(z-2)^2=13

(x+3)^2+(y-1)^2+(z-2)^2=12

(x-3)^2+(y-2)^2+(z-2)^2=52

Correct answer:

(x-3)^2+(y-2)^2+(z-2)^2=13

Explanation:

To find the diameter we need to find the distance between the two points, where:

$D=\sqrt{(6-0)^2+(0-4)^2+(2-2)^2}$

$D=\sqrt{52}=2\sqrt{13}$

Since the radius is half of the Diameter,

$r=\sqrt{13}$

To find the the Center point we use the mid-point formula. So,

$Center = \left ( \frac{0+6}{2},\frac{4+0}{2},\frac{2+2}{2} \right )=\left ( 3,2,2\right )$

So the equation of the sphere is:

(x-3)^2+(y-2)^2+(z-2)^2=13

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #505 : 3 Dimensional Space

Example Question #506 : 3 Dimensional Space

Example Question #507 : 3 Dimensional Space

Example Question #508 : 3 Dimensional Space

Example Question #161 : Spherical Coordinates

Example Question #509 : 3 Dimensional Space

Example Question #166 : Spherical Coordinates

Example Question #167 : Spherical Coordinates

Example Question #168 : Spherical Coordinates

Example Question #511 : 3 Dimensional Space

All Calculus 3 Resources

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