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2 years ago

Twenty-five more than seven times a number is 46 more than four times the number

Mathematics

2 answers:

Marianna [84]2 years ago

6 0

I think the answer is 3

Anarel [89]2 years ago

4 0

Let's say that $x$ is 'the number' we're talking about.

More than means add
Times means multiply
Is means equal to

So all we have to do is set it up in an equation:

$25+7x=46+4x$

So now let's solve for $x$

Get all $x$ s to one side:

$25+3x=46$

Now get all of the numbers to one side:

$3x=24$

Now let's solve for $x$ :

$x=8$

So x=8

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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

8+—3(6+—6a) help please

LenKa [72]

We know that:

$(+)+(-)=(-)$

and:

$(-)*(+)=(-)$

$8+(-3)[6+(-6a)]=8+(-3)*6+(-3)*(-6a)=8-18+18a$ $=(-10)+18a$

if you have solve for $a$ then:

$(-10)+18a=0$
$18a=10$
$a= \frac{10}{18}$
$a=\frac{5}{9}$

5 0

2 years ago

For which function is the domain -6 is less than or equal to x less than or equal to -1 not appropriate? A) y=x^2 B) y=2x C) y=

elena-s [515]

Answer: idk jus needed the points so im answering

3 0

2 years ago

Plz help ASAP I will mark u branliest

-BARSIC- [3]

Answer:

Brainliest?

Step-by-step explanation:

subtract 1 from both sides

divide by 2

add 5 to each side

divide by 4

6 0

2 years ago

Read 2 more answers

Find the area of the figure.

lys-0071 [83]

Answer: 13300

Step-by-step explanation:

3 0

3 years ago