Describe the domain and range of the graph shown.

Is 4:5 (4/5) less, or greater than 19:30 (19/30)?

liberstina [14]

4/5 is greater because when you times it by 6 the numerator becomes 24 therefore it is greater

6 0

3 years ago

Name/ Uid:1. In this problem, try to write the equations of the given surface in the specified coordinates.(a) Write an equation

Gemiola [76]

To find:

(a) Equation for the sphere of radius 5 centered at the origin in cylindrical coordinates

(b) Equation for a cylinder of radius 1 centered at the origin and running parallel to the z-axis in spherical coordinates

Solution:

(a) The equation of a sphere with center at (a, b, c) & having a radius 'p' is given in cartesian coordinates as:

$(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=p^{2}$

Here, it is given that the center of the sphere is at origin, i.e., at (0,0,0) & radius of the sphere is 5. That is, here we have,

$a=b=c=0,p=5$

That is, the equation of the sphere in cartesian coordinates is,

$(x-0)^{2}+(y-0)^{2}+(z-0)^{2}=5^{2}$

$\Rightarrow x^{2}+y^{2}+z^{2}=25$

Now, the cylindrical coordinate system is represented by $(r, \theta,z)$

The relation between cartesian and cylindrical coordinates is given by,

$x=rcos\theta,y=rsin\theta,z=z$

$r^{2}=x^{2}+y^{2},tan\theta=\frac{y}{x},z=z$

Thus, the obtained equation of the sphere in cartesian coordinates can be rewritten in cylindrical coordinates as,

$r^{2}+z^{2}=25$

This is the required equation of the given sphere in cylindrical coordinates.

(b) A cylinder is defined by the circle that gives the top and bottom faces or alternatively, the cross section, & it's axis. A cylinder running parallel to the z-axis has an axis that is parallel to the z-axis. The equation of such a cylinder is given by the equation of the circle of cross-section with the assumption that a point in 3 dimension lying on the cylinder has 'x' & 'y' values satisfying the equation of the circle & that 'z' can be any value.

That is, in cartesian coordinates, the equation of a cylinder running parallel to the z-axis having radius 'p' with center at (a, b) is given by,

$(x-a)^{2}+(y-b)^{2}=p^{2}$

Here, it is given that the center is at origin & radius is 1. That is, here, we have, $a=b=0,p=1$ . Then the equation of the cylinder in cartesian coordinates is,

$x^{2}+y^{2}=1$

Now, the spherical coordinate system is represented by $(\rho,\theta,\phi)$

The relation between cartesian and spherical coordinates is given by,

$x=\rho sin\phi cos\theta,y=\rho sin\phi sin\theta, z= \rho cos\phi$

Thus, the equation of the cylinder can be rewritten in spherical coordinates as,

$(\rho sin\phi cos\theta)^{2}+(\rho sin\phi sin\theta)^{2}=1$

$\Rightarrow \rho^{2} sin^{2}\phi cos^{2}\theta+\rho^{2} sin^{2}\phi sin^{2}\theta=1$

$\Rightarrow \rho^{2} sin^{2}\phi (cos^{2}\theta+sin^{2}\theta)=1$

$\Rightarrow \rho^{2} sin^{2}\phi=1$ (As $sin^{2}\theta+cos^{2}\theta=1$ )

Note that $\rho$ represents the distance of a point from the origin, which is always positive. $\phi$ represents the angle made by the line segment joining the point with z-axis. The range of $\phi$ is given as $0\leq \phi\leq \pi$ . We know that in this range the sine function is positive. Thus, we can say that $sin\phi$ is always positive.

Thus, we can square root both sides and only consider the positive root as,

$\Rightarrow \rho sin\phi=1$

This is the required equation of the cylinder in spherical coordinates.

Final answer:

(a) The equation of the given sphere in cylindrical coordinates is $r^{2}+z^{2}=25$

(b) The equation of the given cylinder in spherical coordinates is $\rho sin\phi=1$

7 0

3 years ago

The LSN basketball team had 292 fans at a game. Adults paid $3.00 to get in and students paid $2.00. A total of $756.00 was coll

rjkz [21]

Answer:

x (378) + y (378) =

Step-by-step explanation:

First i split 756.00 in half so basically what ever is multiplied by two to get 756 and i got 378. So since x = 3.00 and y = 2.00, it would come out to this:

3.00 x 378 = x (378) =

2.00 x 378 = y (378) =

So:

x (378) + y (378) =

im sorry if i got it wrong-

5 0

3 years ago

What are the solutions of the quadratic equation4x*2 − 8x − 12 = 0 ?A) x = −1 and x = −3B) x = −1 and x = 3C) x = 1 and x = −3D)

blsea [12.9K]

The solutions for the given quadratic equation are x = -1 and x = 3. So, option B is correct.

<h3>What are the solutions to an equation?</h3>

The solutions to an equation are the one that satisfies the equation and makes it true.
There are different methods to solve an equation and find its solutions.
These solutions are also called zeros or roots of the equation.

<h3>Calculation:</h3>

The given equation is 4x² - 8x - 12 = 0

Since it is a quadratic equation, it has 2 roots or solutions.

On factorizing the given equation,

4x² - 8x - 12 = 0

⇒ 4x² - 12x + 4x -12 = 0

⇒ 4x(x - 3) + 4(x - 3) = 0

⇒ (4x + 4)(x - 3) = 0

According to the zero-product rule,

4x + 4 = 0 or x - 3 = 0

So, x = -4/4 = -1 or x = 3

Therefore, the solutions of the given equation are -1 and 3.

So, option B) x = -1 and x = 3 is correct.

Learn more about solving an equation here:

brainly.com/question/25678139

#SPJ4

5 0

2 years ago

can somebody please help me with this math swath question and maybe explain how i can work out the answer

Katarina [22]

Answer:

yes a

Step-by-step explanation:

8 0

3 years ago