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3 years ago

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Charles drew a plan for a rectangular piece of material that hehe will use for a blanket. Three of the vertices are ( -2.3 and

−3.6), (−2.3, and 2.2), and (2.1, and 2.2). What are the coordinates of the fourth vertex?

1 answer:

mote1985 [20]3 years ago

8 0

Answer:

The coordinates of the fourth vertex is ( 2.1 , -3.6)

Step-by-step explanation:

Given:

The vertices are ( -2.3 and −3.6), (−2.3, and 2.2), and (2.1, and 2.2).

To Find :

The fourth coordinate = ?

Solution:

Let (a,b) be the missing coordinates

Now let us use the the mid point formula and find the missing coordinate.

Let ( -2.3 ,−3.6) be the point A

Let (−2.3,2.2) be the point B

Let (2.1, 2.2) be the point C

The mid point of AC should be BD

Step 1: Finding The mid pint of AC

The mid pint of AC

$(\frac{(-2.3+2.1)}{2} , \frac{(-3.6+2.2)}{2})$

$(\frac{(-0.2)}{2} , \frac{(-1.4)}{2})$

$(-0.1, -0.7)$

Step 2: Lets Equate the midpoints'

The mid pint of BD

=> $(\frac{(a+(-2.3))}{2} , \frac{(b+2.2)}{2})$ = (-0.1, -0.7)

$\frac{(a+(-2.3))}{2} = -0.1$

$(a-2.3) = -0.1 \times 2$

$(a-2.3) = -0.2$

$a = -0.2 +2.3$

a = 2.1

$\frac{(b+2.2)}{2} = -0.7$

$(b+2.2)= -0.7 \times 2$

$(b+2.2)= -1.4$

$b= -1.4 -2.2$

b = -3.6

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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 $\rho sin\phi=1$

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3 years ago

Ok how is every ones day ok what is 12345 times 23456=

lutik1710 [3]

The answer is 289,564,320. Have a nice day and Happy Holidays

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2 years ago

Write and evaluate the expression. Then, select the correct answer. The sum of forty-five and a number; evaluate when n = 1. 1 4

Colt1911 [192]

The sum of forty-five and a number n is 46.1, and the expression that can represent this can be written "45+n" where the value of n is 1.1

<h3>What is an Expression?</h3>

In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.

The statement that is given to us is" the sum of forty-five and a number", therefore, the expression that will represent this can be written as,

45 + n,

Now, if the value of n is 1.1. Then the sum will be,

$45+n\\\\=45+1.1\\\\=46.1$

Hence, the sum of forty-five and a number n is 46.1, and the expression that can represent this can be written "45+n" where the value of n is 1.1

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2 years ago

Vusi makes bead bracelets in different designs. The first design uses 23 beads, the second design uses 29 beads and the third de

sergey [27]

When the remainder theorem is applied to the total number of beads, the number of beads left is 3

<h3>What is remainder theorem?</h3>

The question is an illustration of remainder theorem. Remainder theorem is used to determine the remainder when a number divides another

The number of beads used in each design are given as:

$First = 23$

$Second = 29$

$Third = 31$

Calculate the total number of beads used for all three designs

$Total =First + Second + Third$

$Total =23 + 29 + 31$

$Total =83$

The number of available beads is:

$Available = 750$

Divide 750 by 83, to get the total number of designs

$n = \frac{750}{83}$

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$n = 9$

The number of beads remaining is calculated using:

$Remaining =Available -n \times Total$

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$Remaining =3$

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2 years ago

Mr. Suárez wants to paint his storage shed. He needs to calculate the lateral surface area of the shed so that he will know how

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Answer:

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Step-by-step explanation:

The picture is the complete question.

The shed is in the shape of a rectangular prism. The lateral surface area of the storage shed can be calculated below. The lateral area is the sides of the prism.

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where

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Lateral surface area of the storage shed = 336 ft²

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2 years ago