3/2t=-1/4

Find the standard equation of a sphere that has diameter with the end points given below. (3,-2,4) (7,12,4)

DiKsa [7]

Answer:

The standard equation of the sphere is $(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = 53$

Step-by-step explanation:

From the question, the end point are (3,-2,4) and (7,12,4)

Since we know the end points of the diameter, we can determine the center (midpoint of the two end points) of the sphere.

The midpoint can be calculated thus

Midpoint = $(\frac{x_{1} + x_{2} }{2}, \frac{y_{1} + y_{2} }{2}, \frac{z_{1} + z_{2} }{2})$

Let the first endpoint be represented as $(x_{1}, y_{1}, z_{1})$ and the second endpoint be $(x_{2}, y_{2}, z_{2})$ .

Hence,

Midpoint = $(\frac{x_{1} + x_{2} }{2}, \frac{y_{1} + y_{2} }{2}, \frac{z_{1} + z_{2} }{2})$

Midpoint = $(\frac{3 + 7 }{2}, \frac{-2+12 }{2}, \frac{4 + 4 }{2})$

Midpoint = $(\frac{10 }{2}, \frac{10}{2}, \frac{8 }{2})\\$

Midpoint = $(5, 5, 4)$

This is the center of the sphere.

Now, we will determine the distance (diameter) of the sphere

The distance is given by

$d = \sqrt{(x_{2} - x_{1})^{2} +(y_{2} - y_{1})^{2} + (z_{2}- z_{1})^{2} }$

$d = \sqrt{(7 - 3)^{2} +(12 - -2)^{2} + (4- 4)^{2}$

$d = \sqrt{(4)^{2} +(14)^{2} + (0)^{2}$

$d = \sqrt{16 +196 + 0$

$d =\sqrt{212}$

$d = 2\sqrt{53}$

This is the diameter

To find the radius, r

From $Radius = \frac{Diameter}{2}$

$Radius = \frac{2\sqrt{53} }{2}$

∴ $Radius = \sqrt{53}$

r = $\sqrt{53}$

Now, we can write the standard equation of the sphere since we know the center and the radius

Center of the sphere is $(5, 5, 4)$

Radius of the sphere is $\sqrt{53}$

The equation of a sphere of radius r and center $(h,k,l)$ is given by

$(x-h)^{2} + (y-k)^{2} + (z-l)^{2} = r^{2}$

Hence, the equation of the sphere of radius $\sqrt{53}$ and center $(5, 5, 4)$ is

$(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = \sqrt{(53} )^{2}$

$(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = 53$

This is the standard equation of the sphere

6 0

3 years ago

(07.06) There are currently 4 people signed up to play on a baseball team. The team must have at least 9 players.

beks73 [17]

A graph which includes the possible values for number of people who can still sign up for the team is: B. number line with closed circle on 5 and shading to the right.

<h3>What is a number line?</h3>

A number line can be defined as a type of graph with a graduated straight line which contains numerical values (positive and negative numbers) that are placed at equal intervals along its length.

Let the number of people who can still sign up for the team be represented by x. Thus, the inequality is given by:

x + 4 ≥ 9

x ≥ 9 - 4

x ≥ 5.

This ultimately implies that, there could be five (5) or more people that can still sign up for the team and a graph which includes these possible values is a number line with closed circle on five (5) and shading to the right because it can get larger.

Read more on number line here: brainly.com/question/24644930

#SPJ1

8 0

1 year ago

The average rainfall of a prairie land is:

bekas [8.4K]

Answer:

Heres the link - brainly.com/question/19882225?referrer=searchResults

Step-by-step explanation:

7 0

3 years ago

Easy question

Ira Lisetskai [31]

Answer:

7/16

Step-by-step explanation:

Let the fraction be x/y

(x+5)/(y+5) = 4/7

7x + 35 = 4y + 20

7x + 15 = 4y

(x-1)/(y-1) = 2/5

5x - 5 = 2y - 2

2y = 5x - 3

7x + 15 = 2(5x - 3)

7x + 15 = 10x - 6

3x = 21

x = 7

2y = 5(7) - 3

2y = 32

y = 16

x/y = 7/16

5 0

3 years ago

What is the surface area of a cylinder? Express the answer in term of x<br><br> D=8cm h=18cm

Ilia_Sergeevich [38]

Answer:

the surface area is 552.92

Step-by-step explanation:

7 0

3 years ago

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