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3 years ago

Solving proportions Solve for k. 10 3 k 9 k

Mathematics

2 answers:

dem82 [27]3 years ago

8 0

Answer:

k = 30

Step-by-step explanation:

Given

$\frac{10}{3}$ = $\frac{k}{9}$ ( cross- multiply )

3k = 90 ( divide both sides by 3 )

k = 30

masya89 [10]3 years ago

8 0

Answer:

K=30

Step-by-step explanation:

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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

A survey at your school found that 842 in 989 students have a cell phone. What percent of the student in school has a cell phone

Mumz [18]

Answer:

85.14% of the students have a cell phone.

8 0

2 years ago

Read 2 more answers

PLS HELP

zhuklara [117]

750.8 square inches

4 0

3 years ago

Margaret’s bottle of shampoo is 7/8 full. She uses 1/3 of the shampoo in the bottle to wash the dog. Estimate what fraction of s

AleksandrR [38]

Fraction of shampoo left = $\frac{13}{24}$

Solution:

Given fraction of full shampoo $=\frac{7}{8}$ .

Fraction of shampoo Margaret used $=\frac{1}{3}$

Fraction of shampoo left = Full shampoo – fraction of shampoo used

$=\frac{7}{8}-\frac{1}{3}$

To make the denominators same do cross multiplication.

$=\frac{7\times3}{8\times3}-\frac{1\times8}{3\times8}$

$=\frac{21}{24}-\frac{8}{24}$

$=\frac{21-8}{24}$

$=\frac{13}{24}$

Hence $\frac{13}{24}$ of the shampoo bottle is left.

8 0

3 years ago

Use the change of base formula to evaluate log3(73).

Marrrta [24]

The change of base formula allows you to write a logarithm in terms of logarithms with another base. It follows this pattern,

$log_{a}x = \frac{ log_{b}x }{log_{b}a }$

where a≠1 and b≠1

Assigning the base to be 7,

$log_{3}73 = \frac{ log_{7}73 }{log_{7}3 }=3.9$

I hope I was able to answer your question. Have a good day.

3 0

3 years ago