1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
raketka [301]
3 years ago
8

Change each of the following points from rectangular coordinates to spherical coordinates and to cylindrical coordinates.

Mathematics
1 answer:
FromTheMoon [43]3 years ago
5 0

Answer and Step-by-step explanation: Spherical coordinate describes a location of a point in space: one distance (ρ) and two angles (Ф,θ).To transform cartesian coordinates into spherical coordinates:

\rho = \sqrt{x^{2}+y^{2}+z^{2}}

\phi = cos^{-1}\frac{z}{\rho}

For angle θ:

  • If x > 0 and y > 0: \theta = tan^{-1}\frac{y}{x};
  • If x < 0: \theta = \pi + tan^{-1}\frac{y}{x};
  • If x > 0 and y < 0: \theta = 2\pi + tan^{-1}\frac{y}{x};

Calculating:

a) (4,2,-4)

\rho = \sqrt{4^{2}+2^{2}+(-4)^{2}} = 6

\phi = cos^{-1}(\frac{-4}{6})

\phi = cos^{-1}(\frac{-2}{3})

For θ, choose 1st option:

\theta = tan^{-1}(\frac{2}{4})

\theta = tan^{-1}(\frac{1}{2})

b) (0,8,15)

\rho = \sqrt{0^{2}+8^{2}+(15)^{2}} = 17

\phi = cos^{-1}(\frac{15}{17})

\theta = tan^{-1}\frac{y}{x}

The angle θ gives a tangent that doesn't exist. Analysing table of sine, cosine and tangent: θ = \frac{\pi}{2}

c) (√2,1,1)

\rho = \sqrt{(\sqrt{2} )^{2}+1^{2}+1^{2}} = 2

\phi = cos^{-1}(\frac{1}{2})

\phi = \frac{\pi}{3}

\theta = tan^{-1}\frac{1}{\sqrt{2} }

d) (−2√3,−2,3)

\rho = \sqrt{(-2\sqrt{3} )^{2}+(-2)^{2}+3^{2}} = 5

\phi = cos^{-1}(\frac{3}{5})

Since x < 0, use 2nd option:

\theta = \pi + tan^{-1}\frac{1}{\sqrt{3} }

\theta = \pi + \frac{\pi}{6}

\theta = \frac{7\pi}{6}

Cilindrical coordinate describes a 3 dimension space: 2 distances (r and z) and 1 angle (θ). To express cartesian coordinates into cilindrical:

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

Angle θ is the same as spherical coordinate;

z = z

Calculating:

a) (4,2,-4)

r=\sqrt{4^{2}+2^{2}} = \sqrt{20}

\theta = tan^{-1}\frac{1}{2}

z = -4

b) (0, 8, 15)

r=\sqrt{0^{2}+8^{2}} = 8

\theta = \frac{\pi}{2}

z = 15

c) (√2,1,1)

r=\sqrt{(\sqrt{2} )^{2}+1^{2}} = \sqrt{3}

\theta = \frac{\pi}{3}

z = 1

d) (−2√3,−2,3)

r=\sqrt{(-2\sqrt{3} )^{2}+(-2)^{2}} = 4

\theta = \frac{7\pi}{6}

z = 3

You might be interested in
The data value x exists in two data sets: A and B. The mean is equal for both data sets. If the standard deviation for set A is
liubo4ka [24]

Answer:

answer is B

it is less than Zx for set B

Step-by-step explanation:

7 0
3 years ago
(-6)*(-6)*(-6)*z*z <br> the expression is
ra1l [238]

Answer:

-216z^2

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
—3x + 1/3y = 12<br> 2y = 18х + 72<br><br> I need the answer ASAP how many solutions and the answer
erica [24]

Answer:

Infinite solutions or all real numbers

Step-by-step explanation:

divide the bottom equation by 2 then....

-3x +1/3 (9x + 36) = 12

-3x + 3x + 12 = 12 (distribute the 1/3 to the terms in the parentheses)

12 = 12

Infinite solutions

8 0
3 years ago
What additional piece of information do we need to prove that MLN ≅ OPN
mote1985 [20]

Answer:

N is the midpoint of MO

Step-by-step explanation:

Given

See attachment for triangles

Required

Additional information to prove MLN \cong OPN

From the attachment, we already have:

LM \simeq OP and LN \simeq ON

The additional information that must be true for MLN \cong OPN  to be true is: <em>N is the midpoint of MO</em>

Because this piece of information proves that:

MN \simeq ON

<em>Which in other words makes </em>MLN \cong OPN<em> </em>

6 0
3 years ago
Only do the ones i have not done brainlest
Leno4ka [110]

Answer:

I don't understand thia

8 0
3 years ago
Read 2 more answers
Other questions:
  • Marlon took a total of 12 pages of notes during 3 hours of class. After attending 4 hours of class, how many total pages of note
    9·1 answer
  • Solve -2(n + 5) + 6n = 14 for n.
    15·1 answer
  • 1 point) during a winter season at one ski resort, there are two roads from area a to area b and two roads from area b to area
    8·1 answer
  • I NEED ANSWER ASAP !!!
    7·1 answer
  • Mia jogs 3 kilometers in 20 minutes. There are about 0.6 miles in a kilometer. What is Mia’s approximate speed in miles per minu
    12·2 answers
  • M is between L and N. LM = 7x-1, MN = 2x +4, and LN = 12. find the value of ‘x’ an determine if M is a bisector.
    6·1 answer
  • Zapisz w jak najprostszej postaci:<br> a) -x-4(1-x)<br> b) 3(5x-2)-5(3-4x)<br> c) √2(3-x)-2(1-x√2)
    9·1 answer
  • 80 timess 40 plus 60
    8·2 answers
  • Solve the equation. − 45 − 5 =
    11·2 answers
  • Yesterday, Natalie's Doughnut Shop sold 3/4 as many chocolate doughnuts as cinnamon doughnuts. If they sold 3/5 of a tray of cin
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!