-4 times a number plus 16 is -116. find the number and show work.

Find the distance between the points (0, 0) and (6, -8). 4 √28 10

kvasek [131]

Answer:

10

Step-by-step explanation:

The formula for finding the distance between 2 points (x1, y1) and (x2, y2) is

d = √[(x2 - x1)² + (y2 - y1)²]

Here (x2, y2) = (6, -8) and (x1, y1) = (0, 0)

Plugging them gives us

d = √[(6 - 0)² + (-8 - 0)²]

d = √[(6)² + (-8)²]

d = √[36 + 64]

d = √100

d = 10

6 0

3 years ago

The volume of this triangular prism is 140 cubic millimeters. What is the value of d? Please need help!!

Lelechka [254]

Answer:

7 millimeters

Step-by-step explanation:

Volume is the area of the base times it's length.

140/10 is 14

so the area of the triangle is 14

the formula for area of a triangle is base times height divided by 2

so

d x 4/ 2 = 14

multiply both sides by 2

d x 4 = 28

divide both sides by 4

d=7

6 0

3 years ago

A bank loaned out $19,000, part of it at the rate of 9 % per year and the rest at 19 % per year. If the interest received in o

VMariaS [17]

Answer: $11100 was loaned at 9%

Step-by-step explanation:

Let x represent the amount that the

bank loaned out at the annual rate of 9%.

Let y represent the amount that the

bank loaned out at the annual rate of 19%.

The bank loaned out $19,000, part of it at the rate of 9% per year and the rest at 19% per year. It means that

x + y = 19000

The interest on $x after 1 year would be

9/100 × x = 0.09x

The interest on $y after 1 year would be

19/100 × y = 0.19y

If the interest received in one year totaled $2500. It means that

0.09x + 0.19y = 2500- - - - - - - - - 1

Substituting x = 19000 - y into equation 1, it becomes

0.09(19000 - y) + 0.19y = 2500

1710 - 19000y + 0.19y = 2500

- 0.09y + 0.19y = 2500 - 1710

- 0.1y = - 790

y = - 790/- 0.1

y = $7900

x = 19000 - y = 19000 - 7900

x = $11100

7 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.